Calculation of Momentum Space Integrals

In the last chapter we have shown that the problem of determining the counterterms in φ 4-theory can be reduced completely to the calculation of massless Feynman integrals in momentum space. Among these, the massless propagator-type integrals generated by the technique of infrared rearrangement in Section 12.2 can be calculated most easily by algebraic methods. The result of a successive application of a generic one-loop integral formula in momentum space yields a Laurent expansion in ε. Some of the integrals can be solved rather directly, others can only be reduced to certain generic two-, three-or four-loop integrals. These can be evaluated by a reduction algorithm in momentum space [1] explained in Section 13.4. In this way, we can find all integrals up to four loops, and most of the five-loop integrals. Some of the five-loop diagrams contain one special type of integrals and a few individual integrals, for which the above reduction algorithm fails. These integrals were initially determined numerically in configuration space, by applying the so-called Gegenbauer-polynomial-x-space technique (GPXT) [2]. Later, however, analytic solutions were found by the method of ideal index constellations [3] described in Section 13.5. With these methods, the pole terms of all Feynman integrals up to five loops have been found analytically. In this chapter we shall be concerned only with the momentum integrals associated with the Feynman diagrams. For this reason, we shall ignore the coupling constants attached to the vertices of the diagrams in the Feynman rule (3.5) of the perturbation expansion. The calculation of all Feynman integrals in momentum space proceeds from the simplest mass-less loop integral d D p (2π) D 1 (p 2) a [(p − k) 2 ] b = q q m k a b. (13.1) For k = 0, this integral vanishes by Veltman's formula (8.33). The powers a and b of the massless propagators will be called line indices. Initially, the line indices of the simple loop diagram are both equal to unity. However, differentiations with respect to the mass or the external momenta discussed in the last chapter, or successive calculations of nested simple loops, will generate indices greater than one. Working in dimensions D = 4 − ε, the line indices will in general be noninteger, but always close to integer with a typical noninteger form a = p + qε/2 , where p and q are integer. The D-dimensional Fourier representation of a …