ar X iv : h ep - p h / 98 02 24 4 v 1 4 F eb 1 99 8 Methods to calculate scalar two - loop vertex diagrams

We present a review of the Bielefeld-Dubna activities on multiloop calculations. In the first contribution of the above authors in these proceedings, we have introduced our system for the automation of evaluation of Feynman diagrams, called DIANA (DIagram ANAlyser). In this contribution methods for the evaluation of scalar two-loop integrals will be discussed. 1 Expansion of three-point functions in terms of external mo-menta squared Taylor series expansions in terms of one external momentum squared, q 2 say, were considered in [1], Padé approximants were introduced in [2] and in Ref. [3] it was demonstrated that this approach can be used to calculate Feynman diagrams on their cut by analytic continuation. The Taylor coefficients are expressed in terms of " bubble diagrams " , i.e. diagrams with external momenta equal zero, which makes their evaluation relatively easy. In the case under consideration we have two independent external momenta in d = 4 − 2ε dimensions. The general expansion of (any loop) scalar 3-point function with its momentum space representation C(p 1 , p 2) can be written as C(p 1 , p 2) = ∞ l,m,n=0 a lmn (p 2 1) l (p 2 2) m (p 1 p 2) n (1) where the coefficients a lmn are to be determined from the given diagram. For many applications it suffices to confine to the case p 2 1 = p 2 2 = 0, which is e.g. physically realized in the case of the Higgs decay into two photons (H → γγ) with p 1 and p 2 the momenta of the photons. In this case only the coefficients a 00n are needed. In the two-loop case we consider the scalar integral (k 3 = k 1 − k 2 , see also Fig. 1)