Analytic treatment of the two loop equal mass sunrise graph Original Research Article

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 image and image, the second order differential equation for the scalar master integral is expanded in image and solved by the variation of the constants method of Euler up to first order in image 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.