Some methods for the evaluation of complicated Feynman integrals

We discuss a progress in calculations of Feynman integrals based on the Gegen-bauer Polynomial Technique and the Differential Equation Method. We demonstrate the results for a class of two-point two-loop diagrams and the evaluation of most complicated part of O(1/N 3) contributions to critical exponents of φ 4-theory. An illustration of the results obtained with help of above methods is considered. Last years there was an essential progress in calculations of Feynman integrals. It seems that most important results have been obtained for two-loop four-point massless Feynman diagrams: in on-shall case (see [1, 2]) and for a class of off-shall legs (see [3]). A review of the results can be found in [4]. Moreover, very recently results for a class of these diagrams have been obtained [5] in the case when some propagators have a nonzero mass. In the paper, I review two methods for calculations of Feynman diagrams. The first one, so-called the Gegenbauer Polynomial Method (see [6] and also [10]-[9]), has been used in particular for the evaluation of α s-corrections to the longitudinal structure function of deep inelastic scattering process. The structure of the results in Mellin moment space (see [10]-[13]) is very similar to the coefficients in [1, 5] of the Mellin-Barnes transforms for the above double-bokses. The coefficients are similar also to ones which have arised (see [14]-[16]) in expansions over the inversed mass for some two-loop two-point and three-point diagrams. A version of the second method, which is called as the Differential Equation Method [17]-[20], has been used in above calculations (see [2, 4] and references therein). An illustration of some results which have been obtained with help of these two methods is considered. The additional information about a modern progress in calculations of Feynman integrals can be found, for example, also in recent articles [21, 22]. Fifteen years ago the method based on the expansion of propagators in Gegenbauer series (see [23]) has been introduced in [6, 7]. One has shown [6, 8] that by this method the analytical evaluation of counterterms in the minimal subtraction scheme at the 4-loop 1