6533b874fe1ef96bd12d6120

RESEARCH PRODUCT

A second-order differential equation for the two-loop sunrise graph with arbitrary masses

Stefan WeinzierlRaphael ZayadehStefan Müller-stach

subject

Loop (graph theory)Algebra and Number TheoryGroup (mathematics)Differential equationMathematical analysisFOS: Physical sciencesGeneral Physics and AstronomyMathematical Physics (math-ph)CohomologyMomentumElliptic curveHigh Energy Physics - PhenomenologyMathematics - Algebraic GeometryHigh Energy Physics - Phenomenology (hep-ph)FOS: MathematicsGraph (abstract data type)Algebraic Geometry (math.AG)Hodge structureMathematical PhysicsMathematics

description

We derive a second-order differential equation for the two-loop sunrise graph in two dimensions with arbitrary masses. The differential equation is obtained by viewing the Feynman integral as a period of a variation of a mixed Hodge structure, where the variation is with respect to the external momentum squared. The fibre is the complement of an elliptic curve. From the fact that the first cohomology group of this elliptic curve is two-dimensional we obtain a second-order differential equation. This is an improvement compared to the usual way of deriving differential equations: Integration-by-parts identities lead only to a coupled system of four first-order differential equations.

yearjournalcountryeditionlanguage
2011-12-19
http://arxiv.org/abs/1112.4360