Numerical evaluation of general massive 2-loop self-mass master integrals from differential equations

Abstract

The system of 4 differential equations in the external invariant satisfied by the 4 master integrals of the general massive 2-loop sunrise self-mass diagram is solved by the Runge-Kutta method in the complex plane. The method offers a reliable and robust approach to the direct and precise numerical evaluation of Feynman graph integrals.The relevance of the higher order calculations for the comparison with nowadays precision measurements in high energy physics is well known and comprehensively presented by G. Passarino in this conference.Therefore can be of some interest the exploitation of an alternative method (but still in the context of the integration by part identities and master integrals (MI) [1]) to the more common direct integration method for the numerical evaluation of the MI.The method uses directly the differential equations. Starting from the integral representation of the MI, related to a certain Feynman graph, by derivation with respect to one of the internal masses [2] or one of the external invariants [3] and with the repeated use of the integration by part identities, a system of independent first order partial differential equations is obtained in a number equal to the number of the MI (master differential equations).Enlarging the number of loops and legs grows the number of parameters, MI and equations, but does not change or spoil the method.

Feynman Diagrams and Differential Equations

“…From that moment on, the method of differential equations became to be widely used in different contexts 11,12,13,14,15,16,17,18,19,20,21,22 . The lists of unprecedented results obtained through its application spans among multi-loop functions from zero to four external legs, 25,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,137,138,139,140,141 and within the most interesting sectors of particle phenomenology, like jets physics 8,9,10,50,51,52,53,54,…”

BOKASUN: A fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams

“…A similar expansion in p 2 around the regular point p 2 = 0 (the first terms where given in [16]) is unpractical due to the severe numerical instability of the coefficients of the expansion [17], associated with the presence of nearby pseudothresholds (this feature is peculiar to the arbitrary mass case, as opposed to the equal mass limit). In the regions were the expansion in 1/p 2 does not work we use the Runge-Kutta algorithm developed in [11,18,19] to obtain the Master Integrals (MI's) of the sunrise diagram as the numerical solutions of a suitable system of linear differential equations. The execution time is of the order of a few seconds (minutes) when the Runge-Kutta method is used (depending on the values of p 2 and the masses and also on the required precision), while it drops to about 80 µs when the new expansion applies and to 3 µs if the expansion coefficients are calculated in advance (all CPU times are given for a 2 GHz PC).…”