Explicit solutions of n - loop vacuum integral recurrence relations

Explicit formulas for solutions of recurrence relations for 3–loop vacuum integrals are generalized for the n-loop case. Recently [1] a new approach to implement recurrence relations [2] for vacuum integrals was suggested. Such relations connect Feynman integrals with various degrees of their denominators. In many cases they provide a possibility to express an integral with given degrees of denominators as a linear combination of a few master integrals with some coefficient functions. The common way to evaluate these functions is step–by–step recurrence procedure, which demands a lot of calculations. On the other hand, the construction of such procedure is a serious problem even at the three-loop level. For vacuum three-loop integrals with one non-zero mass and various numbers of massless lines the corresponding algorithms were constructed in [3]. In the approach proposed in [1] these coefficient functions were calculated directly as solutions of the corresponding recurrence relations. In [1] explicit formulas for the solutions of these relations for 3–loop case with arbitrary masses were obtained. As an example, the case of integrals with four equal masses and two massless lines was considered, and the efficiency of this approach was demonstrated by calculations of the 3-loop QED vacuum polarization. In this work we show how to extend the general formulas for the solutions of the recurrence relations for the multi–loop case. Let us consider L-loop vacuum integrals with N = L(L + 1)/2 denominators (this number of denominators provides the possibility to represent any scalar product of loop momenta as linear combination of the denominators; the diagrams of practical interest which usually have less number of denominators, can be considered as partial cases when some degrees are equal to zero):