Towards a Basis for Planar Two-Loop Integrals

Details Towards a Basis for Planar Two-Loop Integrals Towards a Basis for Planar Two-Loop Integrals

2010-09-02

arxiv.org/abs/1009.0472v2

Phys.Rev.D83:045012,2011

Physics

High Energy Physics

High Energy Physics - Theory

55 pages, 8 figures, Mathematica file for some formulae; final journal version, typos corrected

Scientific paper

10.1103/PhysRevD.83.045012

The existence of a finite basis of algebraically independent one-loop integrals has underpinned important developments in the computation of one-loop amplitudes in field theories and gauge theories in particular. We give an explicit construction reducing integrals to a finite basis for planar integrals at two loops, both to all orders in the dimensional regulator e, and also when all integrals are truncated to O(e). We show how to reorganize integration-by-parts equations to obtain elements of the first basis efficiently, and how to use Gram determinants to obtain additional linear relations reducing this all-orders basis to the second one. The techniques we present should apply to non-planar integrals, to integrals with massive propagators, and beyond two loops as well.

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