FeynGKZ: A Mathematica package for solving Feynman integrals using GKZ hypergeometric systems

Ananthanarayan, B.; Banik, Sumit; Bera, Souvik; Datta, Sudeepan

doi:10.1016/j.cpc.2023.108699

Cited by 24 publications

(10 citation statements)

References 60 publications

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“…Next, we multiply the generators Q i ∈ D by ∂'s from the left, up to a certain degree d max . 3 This yields the equations…”

Section: Pfaffian Systemsmentioning

confidence: 98%

“…There have been various interactions between the relevant mathematics and physics communities already, resulting in several interesting works, for example on determining the number of master integrals [1,9,42], on deriving Picard-Fuchs equations for Feynman integrals [39,43], and on relating Feynman integrals and GKZ systems [3,18,24,37,38,53]. However, to our knowledge, a systematic synthesis and comparison of D-module techniques and methods employed in high energy physics has not yet been done.…”

Section: Introductionmentioning

confidence: 99%

“…Let M be the D n -module corresponding to M. Then, the action of ∂ i on an element m ∈ M = M(X) is given by ∇ ∂ i (m), i.e., ∂ i • m = ∇ ∂ i (m). ⋄ Remark B 3…”

mentioning

confidence: 99%

See 2 more Smart Citations

$D$-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals

Henn¹,

Pratt²,

Sattelberger³

et al. 2023

Preprint

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Feynman integrals are solutions to linear partial differential equations with polynomial coefficients. Using a triangle integral with general exponents as a case in point, we compare D-module methods to dedicated methods developed for solving differential equations appearing in the context of Feynman integrals, and provide a dictionary of the relevant concepts. In particular, we implement an algorithm due to Saito, Sturmfels, and Takayama to derive canonical series solutions of regular holonomic D-ideals, and compare them to asymptotic series derived by the respective Fuchsian systems.

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“…Next, we multiply the generators Q i ∈ D by ∂'s from the left, up to a certain degree d max . 3 This yields the equations…”

Section: Pfaffian Systemsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

$D$-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals

Henn¹,

Pratt²,

Sattelberger³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…For example Regge, in ref. [16], argues, on the basis of homology arguments, that all Feynman integrals must belong to a suitably generalised class of hypergeometric functions, an insight that was sharpened much more recently with the introduction of the Lee-Pomeransky representation [20] of Feynman integrals and the application of the GKZ theory of hypergeometric functions [21][22][23][24][25][26][27][28]. Regge further argues that such functions obey sets of (possibly) high-order differential equations, which he describes as 'a slight generalisation of the well-known Picard-Fuchs equations', also a recurrent theme [29].…”

Section: Jhep03(2024)096mentioning

confidence: 99%

Integration-by-parts identities and differential equations for parametrised Feynman integrals

Artico,

Magnea

2024

J. High Energ. Phys.

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Integration-by-parts (IBP) identities and differential equations are the primary modern tools for the evaluation of high-order Feynman integrals. They are commonly derived and implemented in the momentum-space representation. We provide a different viewpoint on these important tools by working in Feynman-parameter space, and using its projective geometry. Our work is based upon little-known results pre-dating the modern era of loop calculations [16–19, 30, 31]: we adapt and generalise these results, deriving a very general expression for sets of IBP identities in parameter space, associated with a generic Feynman diagram, and valid to any loop order, relying on the characterisation of Feynman-parameter integrands as projective forms. We validate our method by deriving and solving systems of differential equations for several simple diagrams at one and two loops, providing a unified perspective on a number of existing results.

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“…In our previous work, GKZ hypergeometric systems of one-and two-loop Feynman diagrams are obtained [77][78][79] from Mellin-Barnes representations [58,59], through Miller's transformation [80,81]. There are some recent work in GKZ framework of Feynman integrals [82][83][84][85][86][87][88][89][90][91][92][93][94].…”

mentioning

confidence: 99%

GKZ hypergeometric systems of the three-loop vacuum Feynman integrals

Zhang¹,

Feng²

2023

Preprint

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We present the Gel'fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems of the Feynman integrals of the three-loop vacuum diagrams with arbitrary masses, basing on Mellin-Barnes representations and Miller's transformation. The codimension of derived GKZ hypergeometric systems equals the number of independent dimensionless ratios among the virtual masses squared. Through GKZ hypergeometric systems, the analytical hypergeometric series solutions can be obtained in neighborhoods of origin including infinity. The linear independent hypergeometric series solutions whose convergent regions have non-empty intersection can constitute a fundamental solution system in a proper subset of the whole parameter space. The analytical expression of the vacuum integral can be formulated as a linear combination of the corresponding fundamental solution system in certain convergent region.

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FeynGKZ: A Mathematica package for solving Feynman integrals using GKZ hypergeometric systems

Cited by 24 publications

References 60 publications

$D$-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals

$D$-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals

Integration-by-parts identities and differential equations for parametrised Feynman integrals

GKZ hypergeometric systems of the three-loop vacuum Feynman integrals

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