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

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.

[1]  M. Marcolli,et al.  Feynman Integrals and Motives of Configuration Spaces , 2010, 1012.5485.

[2]  M. Marcolli,et al.  ALGEBRO-GEOMETRIC FEYNMAN RULES , 2008, 0811.2514.

[3]  C. Studerus,et al.  Reduze - Feynman integral reduction in C++ , 2009, Comput. Phys. Commun..

[4]  R. N. Lee,et al.  Space-time dimensionality D as complex variable: Calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D , 2009, 0911.0252.

[5]  M. Marcolli.  Feynman integrals and motives , 2009, 0907.0321.

[6]  M. Marcolli,et al.  Motives: an introductory survey for physicists , 2009, 0907.4046.

[7]  M. Marcolli,et al.  Feynman motives and deletion-contraction relations , 2009, 0907.3225.

[8]  M. Marcolli,et al.  Parametric Feynman integrals and determinant hypersurfaces , 2009, 0901.2107.

[9]  Henryk Czyz,et al.  BOKASUN: A fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams , 2008, Comput. Phys. Commun..

[10]  Christian Bogner,et al.  Periods and Feynman integrals , 2007, 0711.4863.

[11]  M. Marcolli,et al.  Feynman motives of banana graphs , 2008, 0807.1690.

[12]  D. Kreimer,et al.  Mixed Hodge Structures and Renormalization in Physics , 2008, 0804.4399.

[13]  P. Mastrolia,et al.  FEYNMAN DIAGRAMS AND DIFFERENTIAL EQUATIONS , 2007, 0707.4037.

[14]  H. Czyz,et al.  Numerical evaluation of the general massive 2-loop sunrise self-mass master integrals from differential equations , 2002, hep-ph/0203256.

[15]  H. Czyz,et al.  The pseudothreshold expansion of the 2-loop sunrise selfmass master amplitudes , 1999, hep-ph/9912501.

[16]  P. Baikov.  Explicit solutions of the multi-loop integral recurrence relations and its application , 1996 .