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 Höschele 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 two-loop five-point integrals. A Mathematica implementation of our algorithm is made available together with this paper.

Highlights

  • Multiple polylogarithms, the weight refers to the number of integrations that are needed to obtain the function, starting from a rational function

  • In subsection 3.2 we present a new result for a four-loop four-point integral and in subsection 3.3 we bring a non-planar four-loop sector that appears in the computation of the angle-dependent cusp-anomalous dimension into canonical form

  • Starting from either a parity-even or parity-odd uniform transcendental weight (UT) integral, the algorithm finds the solution to transform the differential equations on the maximal cut into canonical form, which depend on 17 letters Wi, i ∈ {1, . . . , 5, 11, 16, . . . , 20, 26, . . . , 31}

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Summary

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. G satisfies the canonical differential equations (2.1) with unknown matrix B(x), whose general solution is defined through an iterated integral g(x, ) = P e x x0. With this information, we see that eq (2.16) effectively becomes an equation for the constant ( - and x-independent) matrices ml, and products thereof (projected by the vector v0). Given the assumption that f1 is UT, the Ψ-matrix has rank n and the ansatz for letters αl is complete, the algorithm will terminate at a certain order in , when it finds a non-trivial solution for the ml that determines Φ(x, ) and T (x, ), up to a constant linear transformation. We provide a Mathematica implementation of this algorithm, see section 4

Generalization to multi-variable case
Special cases with degenerate Ψ-matrix and algebraic letters
Examples and applications
Full differential equation for planar three-loop integrals
New result for a four-loop four-point integral
Canonical form for non-planar four-loop sector with 17 master integrals
Four-variable example: non-planar double pentagon integrals
Public implementation
Conclusion and outlook
Full Text
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