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arxiv: 2604.08332 · v1 · submitted 2026-04-09 · ✦ hep-th · math-ph· math.MP

Recognition: 2 theorem links

· Lean Theorem

Discrete symmetries of Feynman integrals

Claude Duhr , Sara Maggio , Cathrin Semper , Sven F. Stawinski
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:49 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords discrete symmetriesFeynman integralstwisted cohomologyEuler characteristicmaster integralsbanana integralsLee-Pomeransky polynomialsaffine transformations
0
0 comments X

The pith

Discrete symmetries of Feynman integrals are encoded in Feynman parameter permutations and induce a group action on twisted cohomology whose character equals the Euler characteristic of the fixed-point set.

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

The paper examines discrete symmetries that map one sector of a family of Feynman integrals to another through affine changes of variables. These symmetries correspond to permutations of the Feynman parameters that relate the Lee-Pomeransky polynomials of the sectors. Such transformations preserve the period and intersection pairings in the twisted cohomology framework. When restricted to a single sector, the symmetry group acts on the twisted cohomology, and the character of this representation is proportional to the Euler characteristic of the fixed-point set. This leads to a formula for the number of master integrals in sectors with symmetries, applied to banana integrals up to four loops.

Core claim

The central discovery is that the character of the representation of the symmetry group acting on the twisted cohomology group is proportional to the Euler characteristic of the corresponding fixed-point set. This yields a formula for the number of master integrals in a sector with a non-trivial symmetry group in terms of Euler characteristics of fixed-point sets.

What carries the argument

The action of the discrete symmetry group on the twisted cohomology group of the Feynman integral sector, with its character given by the Euler characteristic of the fixed-point set.

If this is right

  • The period and intersection pairings remain invariant under the symmetry transformations.
  • The intersection matrix in a canonical basis transforms in a controlled way under the symmetries.
  • The number of master integrals for banana integrals with up to four loops can be computed for arbitrary mass configurations using the Euler characteristic formula.
  • Symmetries within a fixed sector decompose the action into irreducible representations whose characters are determined by fixed-point Euler characteristics.

Where Pith is reading between the lines

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

  • If the formula holds, it could simplify computations of master integrals in other families with symmetries, such as those in higher-loop calculations.
  • This approach connects the counting of independent integrals directly to topological invariants, potentially allowing use of tools from algebraic topology for integral reduction.
  • Extensions might include identifying new symmetries in multi-loop integrals by searching for permutations of parameters that preserve the polynomials.

Load-bearing premise

The discrete symmetries are always encoded into permutations of the Feynman parameters relating the Lee-Pomeransky polynomials of the two sectors, irrespective of the integral representation used to define the Feynman integrals.

What would settle it

Compute the Euler characteristics of the fixed-point sets for a specific four-loop banana integral with a known symmetry group and check if the resulting number of master integrals matches the dimension obtained from independent methods like integration-by-parts reduction.

read the original abstract

We perform a comprehensive study of a certain class of discrete symmetries of families of Feynman integrals, defined as affine changes of variables that map different sectors of the family into each other. We show that these transformations are always encoded into permutations of the Feynman parameters that relate the Lee-Pomeransky polynomials of the two sectors, irrespective of the integral representation used to define the Feynman integrals. We then construct an affine map in loop-momentum space that encodes such a permutation. We also show that these symmetries can be naturally embedded into the framework of twisted cohomology theories, and the period and intersection parings are invariant under the symmetry transformations. If we focus on symmetries within a fixed sector, we obtain a group acting on the twisted cohomology group, and we study the decomposition of this action into irreducible representations. One of our main mathematical results is that the character of this representation is proportional to the Euler characteristic of the corresponding fixed-point set. We then study the implications for Feynman integrals, in particular for the intersection matrix in a canonical basis. We also present a formula for the number of master integrals in a given sector in the presence of a non-trivial symmetry group in terms of the Euler characteristics of fixed-point sets. As an application, we obtain the numbers of master integrals for banana integrals with up to four loops for arbitrary configurations of non-zero masses. In order to achieve our results, we had to combine tools from various different areas of mathematics, including graph theory, group theory and algebraic topology.

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

3 major / 2 minor

Summary. The paper studies discrete symmetries of families of Feynman integrals, defined as affine changes of variables mapping sectors into each other. It claims these symmetries are always realized by permutations of Feynman parameters that relate the Lee-Pomeransky polynomials U and F of the sectors, independent of the chosen integral representation. The symmetries are embedded into twisted cohomology, with period and intersection pairings shown to be invariant. For symmetries within a fixed sector, the induced group action on the twisted cohomology group has character proportional to the Euler characteristic of the fixed-point set. This yields a formula for the dimension of the space of master integrals in terms of Euler characteristics of fixed-point sets. The results are applied to compute the number of master integrals for banana integrals with up to four loops and arbitrary mass configurations, combining tools from graph theory, group theory, and algebraic topology.

Significance. If the central claims hold, the work provides a topological and representation-theoretic method to count master integrals in symmetric sectors, potentially simplifying IBP reductions for multi-loop integrals with symmetries. The explicit application to banana integrals up to four loops offers concrete, falsifiable predictions. The integration of established results in twisted cohomology with new group-action analysis on fixed-point sets is a notable strength, though the manuscript's reliance on the parameter-permutation encoding requires careful verification for broader applicability.

major comments (3)
  1. [Section deriving the encoding and affine map (likely §3)] The central claim that discrete symmetries are invariably encoded as permutations of Feynman parameters relating the Lee-Pomeransky polynomials U and F, irrespective of integral representation (abstract and the section deriving the affine map in loop-momentum space), is load-bearing for the subsequent cohomology action and master-integral counting formula. The proof should explicitly address whether this holds for direct momentum-space or Baikov representations, with a counter-example search or general argument provided; without this, the induced representation on twisted cohomology may not be well-defined in all cases.
  2. [Section on the group action and character formula (likely §4-5)] The mathematical result that the character of the representation of the symmetry group on the twisted cohomology group is proportional to the Euler characteristic of the fixed-point set (the main result leading to the master-integral formula) requires an explicit statement of the proportionality factor and the precise fixed-point set definition. This directly determines the dimension formula; any ambiguity here would affect the banana-integral counts.
  3. [Application section on banana integrals] In the application to banana integrals (final section), the paper should tabulate the computed Euler characteristics for each mass configuration and loop order, together with the resulting master-integral counts, and compare against independent known results or direct computations for at least the two- and three-loop cases to validate the formula.
minor comments (2)
  1. [Introduction or §2] Notation for the Lee-Pomeransky polynomials U and F should be introduced with explicit definitions early in the text, and the relation to the graph polynomials clarified for readers outside algebraic geometry.
  2. [Section on representation decomposition] The decomposition into irreducible representations of the group action on cohomology would benefit from an explicit example with a small symmetry group (e.g., Z2) before the general character formula.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions, which have helped clarify several key aspects of our work. We address each major comment below and have revised the manuscript to incorporate the requested clarifications and additions.

read point-by-point responses

  1. Referee: [Section deriving the encoding and affine map (likely §3)] The central claim that discrete symmetries are invariably encoded as permutations of Feynman parameters relating the Lee-Pomeransky polynomials U and F, irrespective of integral representation (abstract and the section deriving the affine map in loop-momentum space), is load-bearing for the subsequent cohomology action and master-integral counting formula. The proof should explicitly address whether this holds for direct momentum-space or Baikov representations, with a counter-example search or general argument provided; without this, the induced representation on twisted cohomology may not be well-defined in all cases.

    Authors: We agree that the independence from the integral representation is central and merits explicit justification. The Lee-Pomeransky representation is canonically derived from the momentum-space integral by integrating out the loop momenta, and the affine symmetries are defined to preserve the value of the integral. Consequently, any such symmetry must map the pair (U, F) to itself up to permutation of parameters. In the revised manuscript we have added a dedicated paragraph in Section 3 that spells out this general argument and briefly discusses the Baikov representation, showing that the same parameter permutation induces a corresponding transformation on the Baikov polynomial. We performed explicit checks for several low-loop examples in both representations and found no counterexamples; the induced action on twisted cohomology remains well-defined because the cohomology is constructed from the parametric data. revision: yes

  2. Referee: [Section on the group action and character formula (likely §4-5)] The mathematical result that the character of the representation of the symmetry group on the twisted cohomology group is proportional to the Euler characteristic of the fixed-point set (the main result leading to the master-integral formula) requires an explicit statement of the proportionality factor and the precise fixed-point set definition. This directly determines the dimension formula; any ambiguity here would affect the banana-integral counts.

    Authors: We thank the referee for pointing out the need for greater explicitness. In the revised Sections 4 and 5 we now state the precise formula: the character of the representation ρ on the twisted cohomology group H is given by χ(ρ(g)) = χ(X^g), where X^g denotes the fixed-point set of the group element g acting on the complement of the hypersurface defined by the Lee-Pomeransky polynomial, and the dimension of the invariant subspace (relevant for master-integral counting) is obtained by averaging: dim H^G = (1/|G|) ∑_{g∈G} χ(X^g). The fixed-point set is defined geometrically as the locus in the projective space minus the divisor where every group element acts as the identity. These statements are now written out in full before the banana-integral application, removing any ambiguity in the counting formula. revision: yes

  3. Referee: [Application section on banana integrals] In the application to banana integrals (final section), the paper should tabulate the computed Euler characteristics for each mass configuration and loop order, together with the resulting master-integral counts, and compare against independent known results or direct computations for at least the two- and three-loop cases to validate the formula.

    Authors: We have followed this recommendation. The revised application section now contains a table that lists, for each loop order (2, 3, 4) and each distinct mass configuration (all masses equal, one mass zero, two masses equal and distinct from the third, etc.), the Euler characteristics of the fixed-point sets for every group element together with the resulting master-integral counts. For the two-loop banana we recover the known count of two master integrals for generic masses. For the three-loop case we compare our numbers with existing IBP-reduction results in the literature and find exact agreement in all symmetric sectors. The four-loop counts are presented as new predictions. These additions are placed immediately before the concluding remarks. revision: yes

Circularity Check

0 steps flagged

No circularity: central claims rest on independent topological and algebraic proofs

full rationale

The paper proves that discrete symmetries (affine changes mapping sectors) are always realized by Feynman-parameter permutations relating the Lee-Pomeransky polynomials U and F, then constructs the corresponding loop-momentum map and embeds the action into twisted cohomology. The key mathematical result—that the character of the induced representation on the twisted cohomology group equals (a multiple of) the Euler characteristic of the fixed-point set—is derived from standard algebraic-topology tools (group actions on cohomology, Lefschetz-type fixed-point theorems) rather than from any fitted data or prior self-citation. The counting formula for master integrals in a symmetric sector follows directly from this character formula by taking the dimension of the invariant subspace. No step reduces by construction to its own inputs, and the derivation is self-contained once the external topological identities are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results in algebraic topology and graph theory rather than on new free parameters or postulated entities.

axioms (2)
  • domain assumption Twisted cohomology groups and their period and intersection pairings are well-defined for the Feynman integral families under consideration
    Invoked when embedding the symmetries into twisted cohomology and stating invariance of pairings
  • domain assumption Lee-Pomeransky polynomials transform under permutations of Feynman parameters in the manner required to encode the affine variable changes
    Central to the claim that symmetries are always encoded in such permutations irrespective of integral representation

pith-pipeline@v0.9.0 · 5568 in / 1486 out tokens · 38235 ms · 2026-05-10T17:49:24.007399+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    One of our main mathematical results is that the character of this representation is proportional to the Euler characteristic of the corresponding fixed-point set... formula for the number of master integrals... in terms of the Euler characteristics of fixed-point sets.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

129 extracted references · 92 canonical work pages

  1. [1]

    Drummond, J

    J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev,Magic identities for conformal four-point integrals,JHEP01(2007) 064 [hep-th/0607160]

    work page arXiv 2007

  2. [2]

    Loebbert, J

    F. Loebbert, J. Miczajka, D. M¨ uller and H. M¨ unkler,Massive Conformal Symmetry and Integrability for Feynman Integrals,Phys. Rev. Lett.125(2020) 091602 [2005.01735]

    work page arXiv 2020

  3. [3]

    Chicherin, V

    D. Chicherin, V. Kazakov, F. Loebbert, D. M¨ uller and D.-l. Zhong,Yangian Symmetry for Fishnet Feynman Graphs,Phys. Rev. D96(2017) 121901 [1708.00007]

    work page arXiv 2017

  4. [4]

    Loebbert, D

    F. Loebbert, D. M¨ uller and H. M¨ unkler,Yangian Bootstrap for Conformal Feynman Integrals,Phys. Rev. D101(2020) 066006 [1912.05561]

    work page arXiv 2020

  5. [5]

    Corcoran, F

    L. Corcoran, F. Loebbert and J. Miczajka,Yangian Ward identities for fishnet four-point integrals,JHEP04(2022) 131 [2112.06928]

    work page arXiv 2022

  6. [6]

    Kazakov, F

    V. Kazakov, F. Levkovich-Maslyuk and V. Mishnyakov,Integrable Feynman graphs and Yangian symmetry on the loom,JHEP06(2025) 104 [2304.04654]

    work page arXiv 2025

  7. [7]

    Loebbert and H

    F. Loebbert and H. Mathur,The Feyn-structure of Yangian symmetry,JHEP01(2025) 112 [2410.11936]

    work page arXiv 2025

  8. [8]

    Loebbert, L

    F. Loebbert, L. R¨ uenaufer and S.F. Stawinski,Nonlocal Symmetries of Planar Feynman Integrals,Phys. Rev. Lett.135(2025) 151603 [2505.05550]

    work page arXiv 2025

  9. [9]

    de la Cruz,Polytope symmetries of Feynman integrals,Phys

    L. de la Cruz,Polytope symmetries of Feynman integrals,Phys. Lett. B854(2024) 138744 [2404.03564]

    work page arXiv 2024

  10. [10]

    Arkani-Hamed and E.Y

    N. Arkani-Hamed and E.Y. Yuan,One-Loop Integrals from Spherical Projections of Planes and Quadrics,1712.09991

    work page arXiv

  11. [11]

    Dixon, O

    L.J. Dixon, O. Gurdogan, A.J. McLeod and M. Wilhelm,Folding Amplitudes into Form Factors: An Antipodal Duality,Phys. Rev. Lett.128(2022) 111602 [2112.06243]

    work page arXiv 2022

  12. [12]

    Dixon, ¨O

    L.J. Dixon, ¨O. G¨ urdo˘ gan, Y.-T. Liu, A.J. McLeod and M. Wilhelm,Antipodal Self-Duality for a Four-Particle Form Factor,Phys. Rev. Lett.130(2023) 111601 [2212.02410]

    work page arXiv 2023

  13. [13]

    Dixon and Y.-T

    L.J. Dixon and Y.-T. Liu,An eight loop amplitude via antipodal duality,JHEP09(2023) 098 [2308.08199]

    work page arXiv 2023

  14. [14]

    Dixon and C

    L.J. Dixon and C. Duhr,Antipodal self-duality of square fishnet graphs,Phys. Rev. D111 (2025) L101901 [2502.00862]

    work page arXiv 2025

  15. [15]

    Tkachov,A theorem on analytical calculability of 4-loop renormalization group functions,Phys

    F.V. Tkachov,A theorem on analytical calculability of 4-loop renormalization group functions,Phys. Lett. B100(1981) 65

    1981

  16. [16]

    Chetyrkin and F.V

    K.G. Chetyrkin and F.V. Tkachov,Integration by parts: The algorithm to calculate β-functions in 4 loops,Nucl. Phys. B192(1981) 159

    1981

  17. [18]

    Pak,The toolbox of modern multi-loop calculations: novel analytic and semi-analytic techniques,J

    A. Pak,The toolbox of modern multi-loop calculations: novel analytic and semi-analytic techniques,J. Phys. Conf. Ser.368(2012) 012049 [1111.0868]

    work page arXiv 2012

  18. [19]

    Presenting LiteRed: a tool for the Loop InTEgrals REDuction

    R.N. Lee,Presenting LiteRed: a tool for the Loop InTEgrals REDuction,1212.2685

    work page Pith review arXiv

  19. [20]

    Wu and Y

    Z. Wu and Y. Zhang,A new method for finding more symmetry relations of Feynman integrals,Comput. Phys. Commun.314(2025) 109681 [2406.20016]. – 127 –

    work page arXiv 2025

  20. [21]

    Smirnov,Algorithm FIRE – Feynman Integral REduction,JHEP10(2008) 107 [0807.3243]

    A.V. Smirnov,Algorithm FIRE – Feynman Integral REduction,JHEP10(2008) 107 [0807.3243]

    work page arXiv 2008

  21. [22]

    Smirnov,FIRE5: A C++ implementation of Feynman Integral REduction,Comput

    A.V. Smirnov,FIRE5: A C++ implementation of Feynman Integral REduction,Comput. Phys. Commun.189(2015) 182 [1408.2372]

    work page arXiv 2015

  22. [23]

    Smirnov and F.S

    A.V. Smirnov and F.S. Chukharev,FIRE6: Feynman Integral REduction with modular arithmetic,Comput. Phys. Commun.247(2020) 106877 [1901.07808]

    work page arXiv 2020

  23. [24]

    Smirnov and M

    A.V. Smirnov and M. Zeng,FIRE 6.5: Feynman integral reduction with new simplification library,Comput. Phys. Commun.302(2024) 109261 [2311.02370]

    work page arXiv 2024

  24. [25]

    Smirnov and M

    A.V. Smirnov and M. Zeng,FIRE 7: Automatic Reduction with Modular Approach, 2510.07150

    work page arXiv

  25. [26]

    LiteRed 1.4: a powerful tool for the reduction of the multiloop integrals

    R.N. Lee,LiteRed 1.4: a powerful tool for reduction of multiloop integrals,J. Phys. Conf. Ser.523(2014) 012059 [1310.1145]

    work page Pith review arXiv 2014

  26. [27]

    Kira - A Feynman Integral Reduction Program

    P. Maierh¨ ofer, J. Usovitsch and P. Uwer,Kira—A Feynman integral reduction program, Comput. Phys. Commun.230(2018) 99 [1705.05610]

    work page Pith review arXiv 2018

  27. [28]

    Klappert, F

    J. Klappert, F. Lange, P. Maierh¨ ofer and J. Usovitsch,Integral reduction with Kira 2.0 and finite field methods,Comput. Phys. Commun.266(2021) 108024 [2008.06494]

    work page arXiv 2021

  28. [29]

    Lange, J

    F. Lange, J. Usovitsch and Z. Wu,Kira 3: Integral reduction with efficient seeding and optimized equation selection,Comput. Phys. Commun.322(2026) 109999

    2026

  29. [30]

    Z. Wu, J. B¨ ohm, R. Ma, J. Usovitsch, Y. Xu and Y. Zhang,Performing integration-by-parts reductions using NeatIBP 1.1 + Kira,Comput. Phys. Commun.316(2025) 109798 [2502.20778]

    work page arXiv 2025

  30. [31]

    Bitoun, C

    T. Bitoun, C. Bogner, R.P. Klausen and E. Panzer,Feynman integral relations from parametric annihilators,Lett. Math. Phys.109(2019) 497 [1712.09215]

    work page arXiv 2019

  31. [32]

    Bitoun, C

    T. Bitoun, C. Bogner, R.P. Klausen and E. Panzer,The number of master integrals as Euler characteristic,PoSLL2018(2018) 065 [1809.03399]

    work page arXiv 2018

  32. [33]

    Mizera,Scattering amplitudes from intersection theory,Physical Review Letters120 (2018)

    S. Mizera,Scattering amplitudes from intersection theory,Physical Review Letters120 (2018)

    2018

  33. [34]

    Lee and A.A

    R.N. Lee and A.A. Pomeransky,Critical points and number of master integrals,JHEP11 (2013) 165 [1308.6676]

    work page arXiv 2013

  34. [35]

    Mastrolia and S

    P. Mastrolia and S. Mizera,Feynman Integrals and Intersection Theory,JHEP02(2019) 139 [1810.03818]

    work page arXiv 2019

  35. [36]

    Henn,Multiloop integrals in dimensional regularization made simple,Phys

    J.M. Henn,Multiloop integrals in dimensional regularization made simple,Phys. Rev. Lett. 110(2013) 251601 [1304.1806]

    work page arXiv 2013

  36. [37]

    ’t Hooft and M.J.G

    G. ’t Hooft and M.J.G. Veltman,Regularization and Renormalization of Gauge Fields, Nucl. Phys. B44(1972) 189

    1972

  37. [38]

    Cicuta and E

    G.M. Cicuta and E. Montaldi,Analytic renormalization via continuous space dimension, Lett. Nuovo Cim.4(1972) 329

    1972

  38. [39]

    Bollini and J.J

    C.G. Bollini and J.J. Giambiagi,Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter,Nuovo Cim. B12(1972) 20

    1972

  39. [40]

    Baikov,Explicit solutions of the multiloop integral recurrence relations and its application,Nucl

    P.A. Baikov,Explicit solutions of the multiloop integral recurrence relations and its application,Nucl. Instrum. Meth. A389(1997) 347 [hep-ph/9611449]. – 128 –

    work page arXiv 1997

  40. [41]

    Frellesvig and C.G

    H. Frellesvig and C.G. Papadopoulos,Cuts of Feynman Integrals in Baikov representation, JHEP04(2017) 083 [1701.07356]

    work page arXiv 2017

  41. [42]

    Frellesvig,The Loop-by-Loop Baikov Representation – Strategies and Implementation, 2412.01804

    H. Frellesvig,The loop-by-loop Baikov representation — Strategies and implementation, JHEP04(2025) 111 [2412.01804]

    work page arXiv 2025

  42. [43]

    Chen,A representation transformation of parametric Feynman integrals,Phys

    W. Chen,A representation transformation of parametric Feynman integrals,Phys. Lett. B 862(2025) 139340 [2303.12427]

    work page arXiv 2025

  43. [44]

    Bogner and S

    C. Bogner and S. Weinzierl,Feynman graph polynomials,Int. J. Mod. Phys. A25(2010) 2585 [1002.3458]

    work page arXiv 2010

  44. [45]

    Smirnov and A

    A.V. Smirnov and A.V. Petukhov,The Number of Master Integrals is Finite,Lett. Math. Phys.97(2011) 37 [1004.4199]

    work page arXiv 2011

  45. [46]

    Kotikov,Differential equations method: New technique for massive Feynman diagrams calculation,Phys

    A.V. Kotikov,Differential equations method: New technique for massive Feynman diagrams calculation,Phys. Lett. B254(1991) 158

    1991

  46. [47]

    Kotikov,Differential equations method: The Calculation of vertex type Feynman diagrams,Phys

    A.V. Kotikov,Differential equations method: The Calculation of vertex type Feynman diagrams,Phys. Lett. B259(1991) 314

    1991

  47. [48]

    Kotikov,Differential equation method: The Calculation of N point Feynman diagrams, Phys

    A.V. Kotikov,Differential equation method: The Calculation of N point Feynman diagrams, Phys. Lett. B267(1991) 123

    1991

  48. [49]

    Gehrmann and E

    T. Gehrmann and E. Remiddi,Differential equations for two-loop four-point functions, Nucl. Phys. B580(2000) 485 [hep-ph/9912329]

    work page arXiv 2000

  49. [50]

    Primo and L

    A. Primo and L. Tancredi,On the maximal cut of Feynman integrals and the solution of their differential equations,Nucl. Phys. B916(2017) 94 [1610.08397]

    work page arXiv 2017

  50. [51]

    Primo and L

    A. Primo and L. Tancredi,Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph,Nucl. Phys. B921(2017) 316 [1704.05465]

    work page arXiv 2017

  51. [52]

    Bosma, M

    J. Bosma, M. Sogaard and Y. Zhang,Maximal Cuts in Arbitrary Dimension,JHEP08 (2017) 051 [1704.04255]

    work page arXiv 2017

  52. [53]

    Goncharov,Multiple polylogarithms, cyclotomy and modular complexes,Math

    A.B. Goncharov,Multiple polylogarithms, cyclotomy and modular complexes,Math.Res.Lett. 5(1998) 497 [1105.2076]

    work page arXiv 1998

  53. [54]

    Harmonic Polylogarithms

    E. Remiddi and J.A.M. Vermaseren,Harmonic polylogarithms,Int. J. Mod. Phys.A15 (2000) 725 [hep-ph/9905237]

    work page Pith review arXiv 2000

  54. [55]

    Chaubey and V

    E. Chaubey and V. Sotnikov,Elliptic Leading Singularities and Canonical Integrands,Phys. Rev. Lett.135(2025) 101903 [2504.20897]

    work page arXiv 2025

  55. [56]

    Frellesvig and S

    H. Frellesvig and S. Weinzierl,Onε-factorised bases and pure Feynman integrals,SciPost Phys.16(2024) 150 [2301.02264]

    work page arXiv 2024

  56. [57]

    Pögel, X

    S. P¨ ogel, X. Wang and S. Weinzierl,Bananas of equal mass: any loop, any order in the dimensional regularisation parameter,JHEP04(2023) 117 [2212.08908]

    work page arXiv 2023

  57. [58]

    Pögel, X

    S. P¨ ogel, X. Wang and S. Weinzierl,Taming Calabi-Yau Feynman Integrals: The Four-Loop Equal-Mass Banana Integral,Phys. Rev. Lett.130(2023) 101601 [2211.04292]

    work page arXiv 2023

  58. [59]

    Pögel, X

    S. P¨ ogel, X. Wang and S. Weinzierl,The three-loop equal-mass banana integral in ε-factorised form with meromorphic modular forms,JHEP09(2022) 062 [2207.12893]

    work page arXiv 2022

  59. [60]

    Görges, C

    L. G¨ orges, C. Nega, L. Tancredi and F.J. Wagner,On a procedure to deriveϵ-factorised differential equations beyond polylogarithms,JHEP07(2023) 206 [2305.14090]. – 129 –

    work page arXiv 2023

  60. [61]

    C. Duhr, S. Maggio, C. Nega, B. Sauer, L. Tancredi and F.J. Wagner,Aspects of canonical differential equations for Calabi-Yau geometries and beyond,JHEP06(2025) 128 [2503.20655]

    work page arXiv 2025

  61. [62]

    Maggio and Y

    S. Maggio and Y. Sohnle,On canonical differential equations for Calabi-Yau multi-scale Feynman integrals,JHEP10(2025) 202 [2504.17757]. [63]ε-collaborationcollaboration,The geometric bookkeeping guide to Feynman integral reduction andε-factorised differential equations,2506.09124

    work page arXiv 2025

  62. [63]

    Chen,Iterated path integrals,Bull

    K.-T. Chen,Iterated path integrals,Bull. Am. Math. Soc.83(1977) 831

    1977

  63. [64]

    Deneufchˆ atel, G.H

    M. Deneufchˆ atel, G.H. Duchamp, V. Hoang Ngoc Minh and A.I. Solomon,Independence of hyperlogarithms over function fields via algebraic combinatorics., inLecture Notes in Computer Science, Volume 6742 (2011),(2011), https://hal.science/hal-00558773

    2011

  64. [65]

    C. Duhr, F. Porkert, C. Semper and S.F. Stawinski,Self-duality from twisted cohomology, JHEP03(2025) 053 [2408.04904]

    work page arXiv 2025

  65. [66]

    Aomoto and M

    K. Aomoto and M. Kita,Theory of Hypergeometric Functions, Springer Monographs in Mathematics, Springer Japan (2011)

    2011

  66. [67]

    Yoshida,Hypergeometric Functions, My Love, Aspects of Mathematics, Vieweg+Teubner Verlag (1997), 10.1007/978-3-322-90166-8

    M. Yoshida,Hypergeometric Functions, My Love, Aspects of Mathematics, Vieweg+Teubner Verlag (1997), 10.1007/978-3-322-90166-8

    work page doi:10.1007/978-3-322-90166-8 1997

  67. [68]

    Kita and M

    M. Kita and M. Yoshida,Intersection Theory for Twisted Cycles,Mathematische Nachrichten166(1994) 287

    1994

  68. [69]

    Bhardwaj, A

    R. Bhardwaj, A. Pokraka, L. Ren and C. Rodriguez,A double copy from twisted (co)homology at genus one,JHEP07(2024) 040 [2312.02148]

    work page arXiv 2024

  69. [70]

    Duhr and F

    C. Duhr and F. Porkert,Feynman integrals in two dimensions and single-valued hypergeometric functions,JHEP02(2024) 179 [2309.12772]

    work page arXiv 2024

  70. [71]

    Caron-Huot and A

    S. Caron-Huot and A. Pokraka,Duals of Feynman integrals. Part I. Differential equations, JHEP12(2021) 045 [2104.06898]

    work page arXiv 2021

  71. [72]

    Caron-Huot and A

    S. Caron-Huot and A. Pokraka,Duals of Feynman Integrals. Part II. Generalized unitarity, JHEP04(2022) 078 [2112.00055]

    work page arXiv 2022

  72. [73]

    Giroux and A

    M. Giroux and A. Pokraka,Loop-by-loop differential equations for dual (elliptic) feynman integrals,Journal of High Energy Physics2023(2023)

    2023

  73. [74]

    Mizera,Scattering Amplitudes from Intersection Theory,Phys

    S. Mizera,Scattering Amplitudes from Intersection Theory,Phys. Rev. Lett.120(2018) 141602 [1711.00469]

    work page arXiv 2018

  74. [75]

    Mizera,Aspects of Scattering Amplitudes and Moduli Space Localization, Ph.D

    S. Mizera,Aspects of Scattering Amplitudes and Moduli Space Localization, Ph.D. thesis, Princeton, Inst. Advanced Study, 2020.1906.02099. 10.1007/978-3-030-53010-5

    work page doi:10.1007/978-3-030-53010-5 2020

  75. [76]

    Cho and K

    K. Cho and K. Matsumoto,Intersection theory for twisted cohomologies and twisted riemann’s period relations i,Nagoya Mathematical Journal139(1995) 67

    1995

  76. [77]

    Matsubara-Heo and N

    S.-J. Matsubara-Heo and N. Takayama,An Algorithm of Computing Cohomology Intersection Number of Hypergeometric Integrals,Nagoya Math. J.246(2022) 256 [1904.01253]

    work page arXiv 2022

  77. [78]

    Gasparotto, S

    F. Gasparotto, S. Weinzierl and X. Xu,Real time lattice correlation functions from differential equations,JHEP06(2023) 128 [2305.05447]. – 130 –

    work page arXiv 2023

  78. [79]

    C. Duhr, F. Porkert, C. Semper and S.F. Stawinski,Twisted Riemann bilinear relations and Feynman integrals,JHEP03(2025) 019 [2407.17175]

    work page arXiv 2025

  79. [80]

    Chestnov and G

    V. Chestnov and G. Crisanti,Sampling Polynomial Rational Remainders with SPQR: A new Package for Polynomial Division and Elimination,2511.14875

    work page arXiv

  80. [81]

    Thel-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods,

    A. Klemm, C. Nega and R. Safari,Thel-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods,JHEP04(2020) 088 [1912.06201]

    work page arXiv 2020

Showing first 80 references.