3 years ago

All orders structure and efficient computation of linearly reducible elliptic Feynman integrals.

Francesco Moriello, Martijn Hidding

We define linearly reducible elliptic Feynman integrals, and we show that they can be algorithmically solved up to arbitrary order of the dimensional regulator by direct integration of their Feynman parametric representation. We show that the solution, to all orders, can be expressed in terms of a natural generalization of Goncharov polylogarithms, where the last integration kernel is an elementary algebraic function encoding the elliptic nature of the integrals, while all the inner integrations are polylogarithmic. We find empirically that in fact many elliptic Feynman integrals are linearly reducible. We expose these properties by solving representative, previously unknown integrals up to sufficiently high order. The integration algorithm seamlessly applies to single and multiscale integrals. Remarkably, for the examples we consider, it is always possible to perform a basis choice such that the polylogarithmic part of the result is a pure function of uniform transcendental weight. The direct integration method we use is very efficient, as no IBP reductions, boundary conditions, or solution of (higher order) differential equations are required in order to compute a given Feynman integral.

Publisher URL: http://arxiv.org/abs/1712.04441

DOI: arXiv:1712.04441v2

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