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}
Summary
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
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