Belokurov-Usyukina loop reduction in non-integer dimension

TL;DR

This paper extends the Belokurov-Usyukina loop reduction method to non-integer dimensions, enabling exact calculations of certain two-loop diagrams in dimensional regularization without parameter expansion.

Contribution

It generalizes the loop reduction technique to non-integer dimensions, allowing exact evaluation of two-loop diagrams in dimensional regularization.

Findings

Reduction of two-loop triangle diagrams to one-loop diagrams in non-integer dimensions.

Representation of the diagrams in terms of Appell's function F_4.

Exact calculation of diagrams without expansion in the regularization parameter e.

Abstract

Belokurov-Usyukina loop reduction method has been proposed in 1983 to reduce a number of rungs in triangle ladder-like diagram by one. The disadvantage of the method is that it works in d=4 dimensions only and it cannot be used for calculation of amplitudes in field theory in which we are required to put all the incoming and outgoing momenta on shell. We generalize the Belokurov-Usyukina loop reduction technique to non-integer d=4-2e dimensions. In this paper we show how a two-loop triangle diagram with particular values of indices of scalar propagators in the position space can be reduced to a combination of three one-loop scalar diagrams. It is known that any one-loop massless momentum integral can be presented in terms of Appell's function F_4. This means that particular diagram considered in the present paper can be represented in terms of Appell's function F_4 too. Such a generalization of Belokurov-Usyukina loop reduction technique allows us to calculate that diagram by this method exactly without decomposition in terms of the parameter e.