Ivan Gonzalez, Igor Kondrashuk
We construct a class of triangle-ladder diagrams which can be calculated by making a use of Belokurov-Usyukina loop reduction technique in d = 4 -2e dimensions. The main idea of the approach consists in generalization of this loop reduction technique existing in d = 4 dimensions. The recursive formula relating L-loop result for any triangle ladder of that class and (L-1)-loop result for another triangle ladder of the same class is derived. Since the method proposed in the present paper combines analytic and dimensional regularizations, at the end of the calculation we have to remove the analytic regularization by taking the limit in which the parameters of the analytic regularization are vanishing. In this limit on the left hand side of the recursive relations we obtain in the position space the diagram in which the indices of the rungs are 1 and all the other indices are 1-e. Fourier transforms of this type of diagrams are the momentum space diagrams which have indices of the rungs equal to 1-e and all the other indices 1. Via conformal transformation of the dual space image of this momentum space representation we relate such a class of triangle ladder momentum diagrams to a class of the box ladder momentum diagrams in which the indices of the rungs are equal to 1-e and all the other indices are 1. Thus, the proposed generalization of the Belokurov-Usyukina loop reduction technique to non-integer number of dimensions allows us to calculate this class of box-ladder diagrams in the momentum space explicitly in terms of Appell's hypergeometric function F_4 without expanding in powers of parameter e in an arbitrary kinematic region in the momentum space, since any diagram from this class can be reduced to one-loop diagram, including the important on-shell case.
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http://arxiv.org/abs/1210.2243
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