Analytic treatment of the two loop equal mass sunrise graph

Details Analytic treatment of the two loop equal mass sunrise graph Analytic treatment of the two loop equal mass sunrise graph

2004-06-15

arxiv.org/abs/hep-ph/0406160v3

Nucl.Phys. B704 (2005) 349-386

Physics

High Energy Physics

High Energy Physics - Phenomenology

33 pages, LaTeX, v2: +1 figure; v3: changes in the conclusions; simplifications in the recurrences (6.3) and (6.9)

Scientific paper

10.1016/j.nuclphysb.2004.10.044

The two loop equal mass sunrise graph is considered in the continuous d-dimensional regularisation for arbitrary values of the momentum transfer. After recalling the equivalence of the expansions at d=2 and d=4, the second order differential equation for the scalar Master Integral is expanded in (d-2) and solved by the variation of the constants method of Euler up to first order in (d-2) included. That requires the knowledge of the two independent solutions of the associated homogeneous equation, which are found to be related to the complete elliptic integrals of the first kind of suitable arguments. The behaviour and expansions of all the solutions at all the singular points of the equation are exhaustively discussed and written down explicitly.

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