The two-loop sunrise graph with arbitrary masses

We discuss the analytical solution of the two-loop sunrise graph with arbitrary non-zero masses in two space-time dimensions. The analytical result is obtained by solving a second-order differential equation. The solution involves elliptic integrals and in particular the solutions of the corresponding homogeneous differential equation are given by periods of an elliptic curve.

[1]  Phillip A. Griffiths,et al.  On the Periods of Certain Rational Integrals: II , 1969 .

[2]  H. E. Fettis.  On the Reciprocal Modulus Relation for Elliptic Integrals , 1970 .

[3]  F. Tkachov.  A theorem on analytical calculability of 4-loop renormalization group functions , 1981 .

[4]  F. Tkachov,et al.  Integration by parts: The algorithm to calculate β-functions in 4 loops , 1981 .

[5]  Andrei I. Davydychev,et al.  An approach to the evaluation of three- and four-point ladder diagrams , 1993 .

[6]  J. Fleischer,et al.  Two-loop two-point functions with masses: asymptotic expansions and Taylor series, in any dimension , 1993 .

[7]  D. Zagier.  Values of Zeta Functions and Their Applications , 1994 .

[8]  Anthony Joseph,et al.  First European Congress of Mathematics , 1994 .

[9]  M. Böhm,et al.  Closed expressions for specific massive multiloop self-energy integrals , 1994 .

[10]  Analytical and numerical methods for massive two-loop self-energy diagrams , 1994, hep-ph/9409388.

[11]  Calculation of two-loop self-energies in the electroweak Standard Model , 1994, hep-ph/9406404.

[12]  S. Bauberger,et al.  Simple one-dimensional integral representations for two-loop self-energies: the master diagram , 1995 .

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

[14]  Tausk,et al.  Connection between certain massive and massless diagrams. , 1995, Physical review. D, Particles and fields.

[15]  Connection between Feynman integrals having different values of the space-time dimension. , 1996, Physical review. D, Particles and fields.

[16]  O. Tarasov.  Generalized recurrence relations for two-loop propagator integrals with arbitrary masses , 1997, hep-ph/9703319.

[17]  N. I. Ussyukina,et al.  Threshold and pseudothreshold values of the sunset diagram , 1997, hep-ph/9712209.

[18]  H. Czyz,et al.  The Master differential equations for the two loop sunrise selfmass amplitudes , 1998, hep-th/9805118.

[19]  Threshold expansion of the sunset diagram , 1999, hep-ph/9903328.

[20]  S. Groote,et al.  Threshold expansion of Feynman diagrams within a configuration space technique , 2000, hep-ph/0003115.

[21]  The pseudothreshold expansion of the 2-loop sunrise selfmass master amplitudes , 1999, hep-ph/9912501.

[22]  The threshold expansion of the 2-loop sunrise self-mass master amplitudes , 2001, hep-ph/0103014.

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

[24]  A.Onishchenko,et al.  Special case of sunset: reduction and epsilon-expansion , 2002, hep-ph/0207091.

[25]  M. Argeri,et al.  The analytic value of the sunrise self-mass with two equal masses and the external invariant equal to the third squared mass , 2002, hep-ph/0202123.

[26]  The analytical values of the sunrise master integrals for one of the masses equal to zero , 2002, hep-ph/0204039.

[27]  P. Belkale,et al.  Periods and Igusa local zeta functions , 2003 .

[28]  E. Remiddi,et al.  Analytic treatment of the two loop equal mass sunrise graph , 2004, hep-ph/0406160.

[29]  S. Pozzorini,et al.  Precise numerical evaluation of the two loop sunrise graph Master Integrals in the equal mass case , 2006, Comput. Phys. Commun..

[30]  On Motives Associated to Graph Polynomials , 2005, math/0510011.

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

[32]  On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order , 2005, hep-ph/0506286.

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

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

[35]  Jonathan M. Borwein,et al.  Elliptic integral evaluations of Bessel moments and applications , 2008, 0801.0891.

[36]  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..

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

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

[39]  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.

[40]  S. Weinzierl,et al.  A second-order differential equation for the two-loop sunrise graph with arbitrary masses , 2011, 1112.4360.

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

[42]  Marcus Spradlin,et al.  Mellin amplitudes for dual conformal integrals , 2012, 1203.6362.

[43]  S. Groote,et al.  A numerical test of differential equations for one- and two-loop sunrise diagrams using configuration space techniques , 2012, 1204.0694.

[44]  S. Caron-Huot,et al.  Uniqueness of two-loop master contours , 2012, 1205.0801.

[45]  J. Walcher,et al.  Monodromy of an Inhomogeneous Picard-Fuchs Equation , 2012, 1206.1787.

[46]  Stefan Weinzierl,et al.  Picard–Fuchs Equations for Feynman Integrals , 2012, 1212.4389.