Algorithms for minimal Picard–Fuchs operators of Feynman integrals

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.

Section: Jhep03(2024)096mentioning

confidence: 99%

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

Artico,

Magnea

2024

“…[10][11][12][13][14][15][16][17] (see also refs. [18][19][20][21][22][23][24]). The ideals associated with Feynman integrals are holonomic and thus Feynman integrals are holonomic functions [25].…”

Section: Jhep10(2023)098mentioning

confidence: 99%

“…[29] so it would be interesting to construct canonical holonomic representations for these amplitudes and determine whether the resulting differential equations manifest a dependence on sub-amplitudes. (12,3,4,5,6), m 6 (1,23,4,5,6), m 6 (1,2,34,5,6), m 6 (1,2,3,45,6), m 6 (1, 2, 3, 4, 56), m 4 (1, 2, 3)m 5 (123,4,5,6), m 4 (2, 3, 4)m 5 (1,234,5,6), m 4 (3, 4, 5)m 5 (1,2,345,6), respectively, where we have used m 3 (w 1 , w 2 ) = 1.…”

Section: Jhep10(2023)098 4 Conclusionmentioning

confidence: 99%

Holonomic representation of biadjoint scalar amplitudes

de la Cruz

2023

An infinite family of elliptic ladder integrals

McLeod

Morales

Hippel

et al. 2023

We identify two families of ten-point Feynman diagrams that generalize the elliptic double box, and show that they can be expressed in terms of the same class of elliptic multiple polylogarithms to all loop orders. Interestingly, one of these families can also be written as a dlog form. For both families of diagrams, we provide new 2ℓ-fold integral representations that are linearly reducible in all but one variable and that make the above properties manifest. We illustrate the simplicity of this integral representation by directly integrating the three-loop representative of both families of diagrams. These families also satisfy a pair of second-order differential equations, making them ideal examples on which to develop bootstrap techniques involving elliptic symbol letters at high loop orders.