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arxiv: 2606.09978 · v1 · pith:FBVWWRQSnew · submitted 2026-06-08 · ✦ hep-th · hep-ph

Resonance and Differential Reduction of Feynman Integrals

Ruth Britto , Thomas W. Grimm , Arno Hoefnagels This is my paper

Pith reviewed 2026-06-27 15:26 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords Feynman integralsGKZ systemsresonancereduction operatorsdifferential equationsmaster integralsedge contractionone-loop graphs
0
0 comments X

The pith

Resonance in GKZ systems for Feynman integrals yields differential operators that contract graph edges.

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

Feynman integrals appear as solutions to GKZ systems of partial differential equations that typically show resonance. Resonance signals a reducibility to subsystems, which the paper uses to build differential operators that contract edges in the associated Feynman graphs. These operators act compatibly with cuts and close the differential equation systems for a given integral. The remaining GKZ data then supply algebraic relations that shrink the problem to a smaller set reduced to master integrals. The construction is carried out explicitly for one-loop, sunrise, and banana graphs, including cases restricted to physical kinematics, and can combine shifts in propagator powers with spacetime dimension or isolate a pure dimension shift plus edge contraction.

Core claim

Resonance is a type of non-genericity implying reducibility to subsystems. We use this resonance to construct reduction operators, which are differential operators that can contract edges of Feynman graphs. Correspondingly, their action is naturally compatible with cuts of Feynman graphs. Reduction operators may be used to close the system of differential equations for a given integral. The remaining GKZ data lead to algebraic relations identifying a smaller system that is fully reduced to master integrals. We develop the construction for one-loop, sunrise and banana graphs and discuss restrictions to physical kinematics. While reduction operators can generally shift both propagator powers a

What carries the argument

Reduction operators constructed from resonance in GKZ systems, which are differential operators contracting edges of Feynman graphs while remaining compatible with cuts.

If this is right

  • Reduction operators close the differential equation system for a given integral.
  • Remaining GKZ data yield algebraic relations that identify a smaller system reduced to master integrals.
  • Operators can shift both propagator powers and spacetime dimension.
  • Certain operator combinations isolate a pure dimension shift together with contraction of one chosen edge.

Where Pith is reading between the lines

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

  • The same resonance data might generate reduction operators for graphs beyond the one-loop, sunrise, and banana families treated in the paper.
  • Compatibility with cuts could link the operators to cutting rules or unitarity methods in amplitude calculations.
  • The reduction might lower the number of independent integrals needed when restricting to physical kinematics.

Load-bearing premise

Resonance in the GKZ systems for Feynman integrals is a non-genericity that permits explicit construction of edge-contracting reduction operators.

What would settle it

A resonant GKZ system attached to a Feynman graph for which no differential operator constructed from the resonance data contracts a chosen edge.

Figures

Figures reproduced from arXiv: 2606.09978 by Arno Hoefnagels, Ruth Britto, Thomas W. Grimm.

Figure 1
Figure 1. Figure 1: The 1-loop bubble diagram. 4.1 GKZ system for the bubble diagram and resonance The Feynman integral for the bubble diagram, shown in figure 1, takes the fol￾lowing form in the Lee-Pomeransky representation: I D bub(p, m; ν) = κbub(ν) Z R2 + d 2x x ν1−1 1 x ν2−1 2 [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The polytope Conv(A) for the bubble with generic kinematics, with the matrix A given in (63). The integral in equation (62) then satisfies LnI D bub(z; ν) = 0 for n ∈ {1, 2, 3} , (Ee + νe)I D bub(z; ν) = 0 for e ∈ {0, 1, 2} . (66) Resonant facets. We will now show that in the case of generic kinematics, the edge faces are resonant. We have established that there is a face Fe associated to each edge e of th… view at source ↗
Figure 3
Figure 3. Figure 3: Reduction ladder for the one-loop bubble integral [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The polytope Conv(A) for the bubble with m2 = 0, with the matrix A given in (105). 4.5 Some cases of special kinematics We close this section with comments on reduction operators for bubbles with one massless propagator or with equal masses. Bubble with one massless propagator. Consider the bubble depicted in figure 1, now with one of its internal masses set to zero, m2 = 0. The Lee￾Pomeransky polynomial n… view at source ↗
Figure 5
Figure 5. Figure 5: The 1-loop triangle diagram. 5.1 GKZ system and reduction operators The Lee-Pomeransky polynomial for the scalar triangle is G = (x1 + x2 + x3) (1 + m2 1x1 + m2 2x2 + m2 3x3) − p 2 1,2x1x2 − p 2 2,3x2x3 − p 2 1,3x1x3 , (115) which determines the Feynman integral I D tri(p, m; ν) when using (2). In GKZ variables, the integral takes the form I D tri(z; ν) = κtri(ν) Z R3 + d 3x xν1−1 1 x ν2−1 2 x ν3−1 3 G(z, … view at source ↗
Figure 6
Figure 6. Figure 6: The polytope Conv(A) for the triangle with three massive propagators. The column vectors contained in the face F1 are given by a2, a3, a2,3, a2,2, a3,3, and similarly for the other edge faces. Notice that each edge face has the geometry of the polytope of a bubble. with the reduction operator given by Q2,1 = z1∂2 + z1,2∂2,2 + z1,3∂2,3 + 2z1,1∂1,2 , Au = a2, Av = a1 . (124) Here, the subscripts correspond t… view at source ↗
Figure 7
Figure 7. Figure 7: Reduction ladder for the integral I D tri(z; 1, 1, 1) associated to the triangle diagram with ν1 = ν2 = ν3 = 1. First order system and algebraic relations. As in the bubble example, the reduction ladder in figure 7 can be promoted to a system of first-order differential equations. One takes as a preliminary vector ⃗Itri all triangle, bubble and tadpole integrals appearing in the ladder, including the shift… view at source ↗
Figure 8
Figure 8. Figure 8: A general 1-loop n-point diagram. 6.1 GKZ data of one-loop integrals To arrive at the GKZ data, we consider the Lee-Pomeransky polynomials at generic kinematics and with nonvanishing masses me associated to each edge in an n-point diagram. Complexifying the coefficients, the polynomial G(z, x) in the 36 [PITH_FULL_IMAGE:figures/full_fig_p036_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The sunrise diagram. GKZ data and physical locus The GKZ integral for the sunrise diagram is given by I D sun(z; ν) = κsun(ν) Z R3 + d 3x xν1−1 1 x ν2−1 2 x ν3−1 3 G(z, x) −ν0 , (149) where G(z, x) = zˆ1x2x3 + zˆ2x1x3 + zˆ3x1x2 + z0x1x2x3 + z1ˆ2x 2 1x3 + z1ˆ3x 2 1x2 + z2ˆ1x 2 2x3 + z2ˆ3x1x 2 2 + z3ˆ1x2x 2 3 + z3ˆ2x1x 2 3 , (150) 40 [PITH_FULL_IMAGE:figures/full_fig_p040_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The polytope Conv(A) for the sunrise graph. and κsun(ν) is determined from (37) with L = 2, ν = (D/2, ν1, ν2, ν3). For brevity, we have now condensed the indices of the GKZ variables so that they represent the powers of monomials after subtracting one from each. Hats in the indices of the GKZ variables correspond to their absence in their respective monomials. The restriction of the GKZ variables to physi… view at source ↗
Figure 11
Figure 11. Figure 11: The polytopes Conv(A) for sunrise diagrams in the cases that m3 = 0 (left, where one edge face is a facet) and m2 = m3 = 0 (right, where no edge faces are facets) and nonzero masses otherwise. If there are two massless propagators, say m2 2 = m2 3 = 0, then none of the edge faces are facets. There are five facets, none of which are resonant. Therefore, none of the edge faces are resonant. There are no red… view at source ↗
Figure 12
Figure 12. Figure 12: Resonance conditions for the facets of Conv( [PITH_FULL_IMAGE:figures/full_fig_p056_12.png] view at source ↗
read the original abstract

Feynman integrals may be viewed as generalized hypergeometric functions, and specifically as solutions of GKZ systems of partial differential equations that typically exhibit resonance. Resonance is a type of non-genericity implying reducibility to subsystems. We use this resonance to construct reduction operators, which are differential operators that can contract edges of Feynman graphs. Correspondingly, their action is naturally compatible with cuts of Feynman graphs. Reduction operators may be used to close the system of differential equations for a given integral. The remaining GKZ data lead to algebraic relations identifying a smaller system that is fully reduced to master integrals. We develop the construction for one-loop, sunrise and banana graphs and discuss restrictions to physical kinematics. While reduction operators can generally shift both propagator powers and spacetime dimension, certain combinations isolate a pure dimension shift together with contraction of a chosen edge.

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 / 3 minor

Summary. The paper frames Feynman integrals as solutions to GKZ systems of PDEs that generically exhibit resonance, a form of non-genericity that permits reducibility to subsystems. From this resonance the authors construct explicit differential reduction operators that contract edges of the associated Feynman graphs; these operators are compatible with cuts and can be used to close the differential-equation system, after which the remaining GKZ data yield algebraic relations that identify a fully reduced set of master integrals. The construction is carried out in detail for one-loop, sunrise and banana graphs, with additional discussion of restrictions to physical kinematics. Certain linear combinations of the operators are shown to isolate a pure dimension shift together with contraction of a chosen edge.

Significance. If the explicit constructions hold, the work supplies a systematic, resonance-based route to differential reduction operators that is directly tied to the GKZ description of Feynman integrals. The compatibility with cuts and the production of algebraic master-integral relations are practically useful features. The concrete development for the one-loop, sunrise and banana families, together with the kinematic restrictions, provides verifiable examples that can be checked against known results in the literature.

minor comments (3)
  1. The abstract states that 'the remaining GKZ data lead to algebraic relations,' but the manuscript would benefit from an explicit statement, in the main text, of the precise linear combination of GKZ operators that produces each algebraic identity for the sunrise and banana cases.
  2. Notation for the resonance parameters and the resulting reduction operators is introduced without a consolidated table; a short table listing the resonance conditions, the explicit differential operators, and the contracted edges for each graph family would improve readability.
  3. The discussion of physical-kinematics restrictions appears only at the end of each example; moving a brief summary of the allowed kinematic domains to the beginning of each section would help readers assess applicability before the technical construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper frames Feynman integrals as solutions to GKZ systems and uses resonance (a known non-genericity property) to construct reduction operators for specific graphs (one-loop, sunrise, banana). No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described content. The construction is presented as explicit and compatible with cuts, with the remaining GKZ data yielding algebraic relations to master integrals. This aligns with external mathematical structure of GKZ systems rather than reducing to the paper's own inputs by definition. Score 0 is appropriate as the central claim has independent content from the GKZ framework.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper starts from the standard view of Feynman integrals as solutions of GKZ systems; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Feynman integrals may be viewed as generalized hypergeometric functions and specifically as solutions of GKZ systems of partial differential equations that typically exhibit resonance.
    Opening statement of the abstract that frames the entire approach.
  • domain assumption Resonance is a type of non-genericity implying reducibility to subsystems.
    Key property invoked to justify the construction of reduction operators.

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Reference graph

Works this paper leans on

74 extracted references · 23 linked inside Pith

  1. [1]

    Differential Equations for Feynman Integrals,

    P. Vanhove, “Differential Equations for Feynman Integrals,” in International Symposium on Symbolic and Algebraic Computation. 7, 2021

    2021

  2. [2]

    Weinzierl,Feynman Integrals

    S. Weinzierl,Feynman Integrals. A Comprehensive Treatment for Students and Researchers. UNITEXT for Physics. Springer, 2022. arXiv:2201.03593 [hep-th]

    arXiv 2022

  3. [3]

    The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals,

    S. Abreu, R. Britto, and C. Duhr, “The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals,”J. Phys. A55no. 44, (Nov., 2022) 443004,arXiv:2203.13014 [hep-th]. 59

    arXiv 2022

  4. [4]

    The SAGEX review on scattering amplitudes Chapter 4: Multi-loop Feynman integrals,

    J. Bl¨ umlein and C. Schneider, “The SAGEX review on scattering amplitudes Chapter 4: Multi-loop Feynman integrals,”J. Phys. A55 no. 44, (2022) 443005,arXiv:2203.13015 [hep-th]

    arXiv 2022

  5. [5]

    Scattering Amplitudes in Quantum Field Theory,

    S. Badger, J. Henn, J. C. Plefka, and S. Zoia, “Scattering Amplitudes in Quantum Field Theory,”Lect. Notes Phys.1021(2024) pp., arXiv:2306.05976 [hep-th]

    arXiv 2024

  6. [6]

    Critical points and number of master integrals,

    R. N. Lee and A. A. Pomeransky, “Critical points and number of master integrals,”JHEP11(2013) 165,arXiv:1308.6676 [hep-ph]

    Pith/arXiv arXiv 2013

  7. [7]

    Feynman graph polynomials,

    C. Bogner and S. Weinzierl, “Feynman graph polynomials,”Int. J. Mod. Phys. A25(2010) 2585–2618,arXiv:1002.3458 [hep-ph]

    Pith/arXiv arXiv 2010

  8. [8]

    Hypergeometric functions and toric varieties,

    I. M. Gelfand, A. V. Zelevinski˘ ı, and M. M. Kapranov, “Hypergeometric functions and toric varieties,”Funktsional’nyi Analiz i ego Prilozheniya23 no. 2, (1989) 12–26

    1989

  9. [9]

    Generalized Euler integrals and A-hypergeometric functions,

    I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinski˘ ı, “Generalized Euler integrals and A-hypergeometric functions,”Advances in Mathematics84 no. 2, (Dec., 1990) 255–271

    1990

  10. [10]

    I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinski˘ ı,Discriminants, Resultants, and Multidimensional Determinants. Mathematics. Birkh¨ auser, Boston Basel Berlin, 1994

    1994

  11. [11]

    Saito, B

    M. Saito, B. Sturmfels, and N. Takayama,Gr¨ obner Deformations of Hypergeometric Differential Equations, vol. 6 ofAlgorithms and Computation in Mathematics. Springer, Berlin, Heidelberg, 2000

    2000

  12. [12]

    Feynman integrals, toric geometry and mirror symmetry,

    P. Vanhove, “Feynman integrals, toric geometry and mirror symmetry,” in KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, pp. 415–458. 2019.arXiv:1807.11466 [hep-th]

    Pith/arXiv arXiv 2019

  13. [13]

    Feynman integrals as A-hypergeometric functions,

    L. de la Cruz, “Feynman integrals as A-hypergeometric functions,”JHEP 12(2019) 123,arXiv:1907.00507 [math-ph]

    arXiv 2019

  14. [14]

    Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems,

    R. P. Klausen, “Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems,”Journal of High Energy Physics2020no. 4, (Apr., 2020) 121,arXiv:1910.08651

    arXiv 2020

  15. [15]

    FeynGKZ: A Mathematicapackage for solving Feynman integrals using GKZ hypergeometric systems,

    B. Ananthanarayan, S. Banik, S. Bera, and S. Datta, “FeynGKZ: A Mathematicapackage for solving Feynman integrals using GKZ hypergeometric systems,”Comput. Phys. Commun.287(Feb., 2023) 108699,arXiv:2211.01285 [hep-th]. 60

    arXiv 2023

  16. [16]

    Cohen-Macaulay Property of Feynman Integrals,

    F. Tellander and M. Helmer, “Cohen-Macaulay Property of Feynman Integrals,”Communications in Mathematical Physics399no. 2, (Apr.,

  17. [17]

    On Feynman graphs, matroids, and GKZ-systems,

    U. Walther, “On Feynman graphs, matroids, and GKZ-systems,”Lett. Math. Phys.112no. 6, (2022) 120,arXiv:2206.05378 [math-ph]

    arXiv 2022

  18. [18]

    Characterizing Cohen-Macaulay One-Loop Feynman Integrals,

    K. Michaelsen and F. Tellander, “Characterizing Cohen-Macaulay One-Loop Feynman Integrals,”arXiv:2512.13820 [hep-th]

    arXiv

  19. [19]

    Isomorphism Classes of A-Hypergeometric Systems,

    M. Saito, “Isomorphism Classes of A-Hypergeometric Systems,” Compositio Mathematica128no. 3, (2001) 323–338,math/9912213

    Pith/arXiv arXiv 2001

  20. [20]

    Irreducibility of A-hypergeometric systems,

    F. Beukers, “Irreducibility of A-hypergeometric systems,”Indagationes Mathematicae21no. 1-2, (July, 2011) 30–39

    2011

  21. [21]

    Algebraic aspects of hypergeometric differential equations,

    T. Reichelt, M. Schulze, C. Sevenheck, and U. Walther, “Algebraic aspects of hypergeometric differential equations,”Beitr¨ age zur Algebra und Geometrie / Contributions to Algebra and Geometry62no. 1, (Mar., 2021) 137–203

    2021

  22. [22]

    Dwork,Generalized Hypergeometric Functions

    B. Dwork,Generalized Hypergeometric Functions. Oxford Mathematical Monographs. Oxford University Press, Oxford, New York, Aug., 1990

    1990

  23. [23]

    A-hypergeometric functions and creation operators for Feynman and Witten diagrams,

    F. Caloro and P. McFadden, “A-hypergeometric functions and creation operators for Feynman and Witten diagrams,” Sept., 2023. arXiv:2309.15895

    arXiv 2023

  24. [24]

    Reductions of GKZ systems and applications to cosmological correlators,

    T. W. Grimm and A. Hoefnagels, “Reductions of GKZ systems and applications to cosmological correlators,”JHEP04(2025) 196, arXiv:2409.13815 [hep-th]

    arXiv 2025

  25. [25]

    A reduction algorithm for cosmological correlators: cuts, contractions, and complexity,

    T. W. Grimm, A. Hoefnagels, and M. van Vliet, “A reduction algorithm for cosmological correlators: cuts, contractions, and complexity,”JHEP03 (2026) 208,arXiv:2503.05866 [hep-th]

    arXiv 2026

  26. [26]

    Holonomic Character and Local Monodromy Structure of Feynman Integrals,

    M. Kashiwara and T. Kawai, “Holonomic Character and Local Monodromy Structure of Feynman Integrals,”Commun. Math. Phys.54(1977) 121–134

    1977

  27. [27]

    Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces,

    S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, “Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces,”Commun. Math. Phys.167no. 2, (1995) 301–350,arXiv:hep-th/9308122

    Pith/arXiv arXiv 1995

  28. [28]

    GKZ generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces,

    S. Hosono, B. H. Lian, and S.-T. Yau, “GKZ generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces,”Commun. Math. Phys.182(1996) 535–578. 61

    1996

  29. [29]

    GKZ hypergeometric systems and applications to mirror symmetry,

    S. Hosono and B. H. Lian, “GKZ hypergeometric systems and applications to mirror symmetry,” inFrontiers in Quantum Field Theory in Honor of the 60th Birthday of Prof. K. Kikkawa, pp. 86–95. Feb., 1996. arXiv:hep-th/9602147

    Pith/arXiv arXiv 1996

  30. [30]

    A Feynman integral via higher normal functions,

    S. Bloch, M. Kerr, and P. Vanhove, “A Feynman integral via higher normal functions,”Compos. Math.151no. 12, (2015) 2329–2375, arXiv:1406.2664 [hep-th]

    Pith/arXiv arXiv 2015

  31. [31]

    Local mirror symmetry and the sunset Feynman integral,

    S. Bloch, M. Kerr, and P. Vanhove, “Local mirror symmetry and the sunset Feynman integral,”Adv. Theor. Math. Phys.21(2017) 1373–1453, arXiv:1601.08181 [hep-th]

    Pith/arXiv arXiv 2017

  32. [32]

    Picard-Fuchs equations for Feynman integrals,

    S. M¨ uller-Stach, S. Weinzierl, and R. Zayadeh, “Picard-Fuchs equations for Feynman integrals,”Communications in Mathematical Physics326no. 1, (Feb., 2014) 237–249,arXiv:1212.4389 [hep-ph, physics:math-ph]

    Pith/arXiv arXiv 2014

  33. [33]

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

    A. Klemm, C. Nega, and R. Safari, “The L-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods,”JHEP04no. 4, (Apr.,

  34. [34]

    088,arXiv:1912.06201 [hep-th]

    arXiv 1912

  35. [35]

    Analytic structure of all loop banana integrals,

    K. B¨ onisch, F. Fischbach, A. Klemm, C. Nega, and R. Safari, “Analytic structure of all loop banana integrals,”JHEP05(May, 2021) 066, arXiv:2008.10574 [hep-th]

    arXiv 2021

  36. [36]

    Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives,

    K. B¨ onisch, C. Duhr, F. Fischbach, A. Klemm, and C. Nega, “Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives,”JHEP09(Sept., 2022) 156,arXiv:2108.05310 [hep-th]

    arXiv 2022

  37. [37]

    Algorithms for minimal Picard–Fuchs operators of Feynman integrals,

    P. Lairez and P. Vanhove, “Algorithms for minimal Picard–Fuchs operators of Feynman integrals,”Lett. Math. Phys.113no. 2, (Mar., 2023) 37, arXiv:2209.10962 [hep-th]

    arXiv 2023

  38. [38]

    Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case,

    V. V. Bytev, M. Y. Kalmykov, and B. A. Kniehl, “Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case,”Nucl. Phys. B836(2010) 129–170,arXiv:0904.0214 [hep-th]

    Pith/arXiv arXiv 2010

  39. [39]

    Counting master integrals: Integration by parts versus differential reduction,

    M. Y. Kalmykov and B. A. Kniehl, “Counting master integrals: Integration by parts versus differential reduction,”Phys. Lett. B702(2011) 268–271, arXiv:1105.5319 [math-ph]

    Pith/arXiv arXiv 2011

  40. [40]

    Mellin-Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions,

    M. Y. Kalmykov and B. A. Kniehl, “Mellin-Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions,”Phys. Lett. B714(2012) 103–109,arXiv:1205.1697 [hep-th]. 62

    Pith/arXiv arXiv 2012

  41. [41]

    Specializations of partial differential equations for Feynman integrals,

    V. V. Bytev, B. A. Kniehl, and O. L. Veretin, “Specializations of partial differential equations for Feynman integrals,”Nucl. Phys. B984(2022) 115972,arXiv:2207.08565 [hep-th]

    arXiv 2022

  42. [42]

    Gauss relations in Feynman integrals,

    T.-F. Feng, Y. Zhou, and H.-B. Zhang, “Gauss relations in Feynman integrals,”Phys. Rev. D111no. 1, (2025) 016015,arXiv:2407.10287 [hep-th]

    arXiv 2025

  43. [43]

    Feynman integral relations from parametric annihilators,

    T. Bitoun, C. Bogner, R. P. Klausen, and E. Panzer, “Feynman integral relations from parametric annihilators,”Lett. Math. Phys.109no. 3, (Aug., 2018) 497–564,arXiv:1712.09215 [hep-th]

    Pith/arXiv arXiv 2018

  44. [44]

    CALICO: Computing Annihilators from Linear Identities Constraining (differential) Operators,

    G. Bertolini, G. Fontana, and T. Peraro, “CALICO: Computing Annihilators from Linear Identities Constraining (differential) Operators,” JHEP10(2025) 018,arXiv:2506.13653 [hep-ph]

    arXiv 2025

  45. [45]

    Macaulay matrix for Feynman integrals: Linear relations and intersection numbers,

    V. Chestnov, F. Gasparotto, M. K. Mandal, P. Mastrolia, S. J. Matsubara-Heo, H. J. Munch, and N. Takayama, “Macaulay matrix for Feynman integrals: Linear relations and intersection numbers,”JHEP09 (2022) 187,arXiv:2204.12983 [hep-th]

    arXiv 2022

  46. [46]

    Restrictions of Pfaffian systems for Feynman integrals,

    V. Chestnov, S. J. Matsubara-Heo, H. J. Munch, and N. Takayama, “Restrictions of Pfaffian systems for Feynman integrals,”JHEP11(2023) 202,arXiv:2305.01585 [hep-th]

    arXiv 2023

  47. [47]

    Vector Spaces of Generalized Euler Integrals,

    D. Agostini, C. Fevola, A.-L. Sattelberger, and S. Telen, “Vector Spaces of Generalized Euler Integrals,”Commun. Num. Theor. Phys.18no. 2, (2024) 327–370,arXiv:2208.08967 [math.AG]

    arXiv 2024

  48. [48]

    Differential space of Feynman integrals: Annihilators andD-module,

    V. Chestnov, W. Flieger, P. Mastrolia, S.-J. Matsubara-Heo, N. Takayama, and W. J. Torres Bobadilla, “Differential space of Feynman integrals: Annihilators andD-module,”Phys. Lett. B871(2025) 140023, arXiv:2506.10456 [hep-th]

    arXiv 2025

  49. [49]

    Picard–fuchs equations of twisted differential forms associated to Feynman integrals,

    P. Vanhove, “Picard–fuchs equations of twisted differential forms associated to Feynman integrals,” 2026. arXiv:2604.09129; proceedings contribution to Regulators V

    Pith/arXiv arXiv 2026

  50. [50]

    Britto, C

    R. Britto, C. Duhr, H. S. Hannesdottir, and S. Mizera,Cutting-Edge Tools for Cutting Edges, p. 595–620. Elsevier, 2025.arXiv:2402.19415

    arXiv 2025

  51. [51]

    Restriction of hypergeometricD-modules with respect to coordinate subspaces,

    M.-C. Fern´ andez-Fern´ andez and U. Walther, “Restriction of hypergeometricD-modules with respect to coordinate subspaces,” Proceedings of the American Mathematical Society139no. 9, (2011) 3175–3180. 63

    2011

  52. [52]

    Principal Landau determinants,

    C. Fevola, S. Mizera, and S. Telen, “Principal Landau determinants,” Comput. Phys. Commun.303(2024) 109278,arXiv:2311.16219 [math-ph]

    arXiv 2024

  53. [53]

    Feynman Amplitudes and Cosmic Galois group,

    F. Brown, “Feynman Amplitudes and Cosmic Galois group,”Commun. Num. Theor. Phys.11(2017) 453–556,arXiv:1512.06409 [math-ph]

    Pith/arXiv arXiv 2017

  54. [54]

    Singularities and discontinuities of Feynman amplitudes,

    R. E. Cutkosky, “Singularities and discontinuities of Feynman amplitudes,” J. Math. Phys.1(1960) 429–433

    1960

  55. [55]

    Generalized Cuts of Feynman Integrals in Parameter Space,

    R. Britto, “Generalized Cuts of Feynman Integrals in Parameter Space,” Phys. Rev. Lett.131no. 9, (Sept., 2023) 091601,arXiv:2305.15369 [hep-th]

    arXiv 2023

  56. [56]

    Euler-Mellin integrals and A-hypergeometric functions,

    C. Berkesch, J. Forsg˚ ard, and M. Passare, “Euler-Mellin integrals and A-hypergeometric functions,”Michigan Math. J.63no. 1, (Mar., 2014) 101–123,arXiv:1103.6273 [math.CV]

    Pith/arXiv arXiv 2014

  57. [57]

    Hoefnagels,Differential Reductions and Cosmological Correlations

    A. Hoefnagels,Differential Reductions and Cosmological Correlations. Thesis, Universiteit Utrecht, October, 2025.2512.19775 [hep-th]

    arXiv 2025

  58. [58]

    Incomplete A-hypergeometric systems,

    K. Nishiyama and N. Takayama, “Incomplete A-hypergeometric systems,” inHarmony of Gr¨ obner Bases and the Modern Industrial Society, pp. 193–212. World Scientific, Hackensack, NJ, 2012.arXiv:0907.0745 [math.CA]

    Pith/arXiv arXiv 2012

  59. [59]

    Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces,

    S. Li, B. H. Lian, and S.-T. Yau, “Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces,”Am. J. Math. 134no. 5, (2012) 1345–1384,arXiv:0910.4215 [math.AG]

    Pith/arXiv arXiv 2012

  60. [60]

    Duality and monodromy reducibility of A-hypergeometric systems,

    U. Walther, “Duality and monodromy reducibility of A-hypergeometric systems,”Mathematische Annalen338(2005) 55–74, arXiv:math/0508622

    Pith/arXiv arXiv 2005

  61. [61]

    Laurent Polynomials, GKZ-hypergeometric Systems and Mixed Hodge Modules,

    T. Reichelt, “Laurent Polynomials, GKZ-hypergeometric Systems and Mixed Hodge Modules,”Compositio Mathematica150no. 6, (June, 2014) 911–941,arXiv:1209.3941 [math]

    Pith/arXiv arXiv 2014

  62. [62]

    Gauss’ hypergeometric function,

    F. Beukers, “Gauss’ hypergeometric function,” inArithmetic and Geometry Around Hypergeometric Functions, vol. 260 ofProgress in Mathematics, pp. 23–42. Birkh¨ auser, Basel, 2007

    2007

  63. [63]

    Hypergeometric Hodge modules,

    T. Reichelt and C. Sevenheck, “Hypergeometric Hodge modules,” arXiv:1503.01004 [math]

    arXiv

  64. [64]

    Three lectures on hypergeometric functions,

    E. Cattani, “Three lectures on hypergeometric functions,”Notes for a course(2006) . 64

    2006

  65. [65]

    Irreducible quotients of A-hypergeometric systems,

    M. Saito, “Irreducible quotients of A-hypergeometric systems,”Compositio Mathematica147no. 2, (Mar., 2011) 613–632

    2011

  66. [66]

    Resonance equals reducibility for A-hypergeometric systems,

    M. Schulze and U. Walther, “Resonance equals reducibility for A-hypergeometric systems,”Algebra & Number Theory6no. 3, (July,

  67. [67]

    Resonant Contractions of One-Loop Feynman Integrals

    A. Hoefnagels and F. Tellander, “Resonant Contractions of One-Loop Feynman Integrals.” In preparation

  68. [68]

    Multiloop integrals in dimensional regularization made simple,

    J. M. Henn, “Multiloop integrals in dimensional regularization made simple,”Phys. Rev. Lett.110(June, 2013) 251601,arXiv:1304.1806 [hep-th]

    Pith/arXiv arXiv 2013

  69. [69]

    Cuts from residues: the one-loop case,

    S. Abreu, R. Britto, C. Duhr, and E. Gardi, “Cuts from residues: the one-loop case,”JHEP06(2017) 114,arXiv:1702.03163 [hep-th]

    Pith/arXiv arXiv 2017

  70. [70]

    The diagrammatic coaction beyond one loop,

    S. Abreu, R. Britto, C. Duhr, E. Gardi, and J. Matthew, “The diagrammatic coaction beyond one loop,”JHEP10(2021) 131, arXiv:2106.01280 [hep-th]

    arXiv 2021

  71. [71]

    Complexity of computations with Pfaffian and Noetherian functions,

    A. Gabrielov and N. Vorobjov, “Complexity of computations with Pfaffian and Noetherian functions,” inNormal forms, bifurcations and finiteness problems in differential equations, vol. 137 ofNATO Sci. Ser. II Math. Phys. Chem., pp. 211–250. Kluwer Acad. Publ., Dordrecht, 2004

    2004

  72. [72]

    Sharply o-minimal structures and sharp cellular decomposition,

    G. Binyamini, D. Novikov, and B. Zack, “Sharply o-minimal structures and sharp cellular decomposition,”arXiv:2209.10972 [math]

    arXiv

  73. [73]

    On the complexity of quantum field theory,

    T. W. Grimm and M. van Vliet, “On the complexity of quantum field theory,”JHEP06(2025) 215,arXiv:2410.23338 [hep-th]

    arXiv 2025

  74. [74]

    GKZ hypergeometric structures,

    J. Stienstra, “GKZ hypergeometric structures,” inInstanbul 2005: CIMPA Summer School on Arithmetic and Geometry Around Hypergeometric Functions. 11, 2005.arXiv:math/0511351. 65

    Pith/arXiv arXiv 2005