Deriving canonical differential equations for Feynman integrals from a single uniform weight integral
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Springer
Received: March 11, 2020 Accepted: April 18, 2020 Published: May 7, 2020
Christoph Dlapa, Johannes Henn and Kai Yan Max-Planck-Institut f¨ ur Physik, Werner-Heisenberg-Institut, D-80805 M¨ unchen, Germany
E-mail: [email protected], [email protected], [email protected] Abstract: Differential equations are a powerful tool for evaluating Feynman integrals. Their solution is straightforward if a transformation to a canonical form is found. In this paper, we present an algorithm for finding such a transformation. This novel technique is based on a method due to H¨ oschele et al. and relies only on the knowledge of a single integral of uniform transcendental weight. As a corollary, the algorithm can also be used to test the uniform transcendentality of a given integral. We discuss the application to several cutting-edge examples, including non-planar four-loop HQET and non-planar twoloop five-point integrals. A Mathematica implementation of our algorithm is made available together with this paper. Keywords: NLO Computations ArXiv ePrint: 2002.02340
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)025
JHEP05(2020)025
Deriving canonical differential equations for Feynman integrals from a single uniform weight integral
Contents 1
2 Description of the method 2.1 From the Picard-Fuchs equation of a uniform weight Feynman integral to a canonical system of first-order equations 2.2 Ansatz for canonical differential equations, and determination of unknowns 2.3 Generalization to multi-variable case 2.4 Special cases with degenerate Ψ-matrix and algebraic letters
3 3 6 8 10
3 Examples and applications 3.1 Full differential equation for planar three-loop integrals 3.2 New result for a four-loop four-point integral 3.3 Canonical form for non-planar four-loop sector with 17 master integrals 3.4 Four-variable example: non-planar double pentagon integrals
13 13 14 16 17
4 Public implementation
19
5 Conclusion and outlook
19
1
Introduction
Feynman integrals are ubiquitous in perturbative quantum field theory. They are required in order to extract predictions from the theory beyond the leading perturbative order. As a consequence, they are important in many areas. A prominent example is collider physics, where the underlying scattering processes are described by Feynman diagrams, and consequently on-shell momentum space Feynman integrals are needed. Another example are off-shell position-space correlation functions (e.g. in conformal field theory), from which one can determine the scaling dimension of fields, or renormalization group coefficients. Beyond their obvious importance for physics, Feynman integrals also have interesting connections to mathematics. The reason is that Feynman integrals are periods [1, 2], and give rise to interesting classes of special functions. Moreover, they can be studied using partial differential equations, and algebraic geometry plays an important role in the analysis of the integrands. Given these motivations, it is not surprising that thi
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