Box ladders in a noninteger dimension

We construct a family of triangle-ladder diagrams that can be calculated using the Belokurov–Usyukina loop reduction technique in $d=4-2\varepsilon$ dimensions. The main idea of the approach we propose is to generalize this loop reduction technique existing in $d=4$ dimensions. We derive a recurrence relation between the result for an $L$-loop triangle-ladder diagram of this family and the result for an $(L-1)$-loop triangle-ladder diagram of the same family. Because the proposed method combines analytic and dimensional regularizations, we must remove the analytic regularization at the end of the calculation by taking the double uniform limit in which the parameters of the analytic regularization vanish. In the position space, we obtain a diagram in the left-hand side of the recurrence relations in which the rung indices are $1$ and all other indices are $1-\varepsilon$ in this limit. Fourier transforms of diagrams of this type give momentum space diagrams with rung indices $1-\varepsilon$ and all other indices $1$. By a conformal transformation of the dual space image of this momentum space representation, we relate such a family of triangle-ladder momentum diagrams to a family of box-ladder momentum diagrams with rung indices $1-\varepsilon$ and all other indices $1$. Because any diagram from this family is reducible to a one-loop diagram, the proposed generalization of the Belokurov–Usyukina loop reduction technique to a noninteger number of dimensions allows calculating this family of box-ladder diagrams in the momentum space explicitly in terms of Appell's hypergeometric function $F_4$ without expanding in powers of the parameter $\varepsilon$ in an arbitrary kinematic region in the momentum space.