Method of regions for dual conformal integrals

Abstract

We apply the method of regions to the evaluation of dual conformal integrals with small off-shellness. In contrast to conventional approach, where the separation of regions is performed via dimensional regularization breaking the dual conformal invariance (DCI), we use a sufficiently generic combination of dimensional and analytic regularizations which preserves the DCI. Within this regularization (dubbed as DCI regularization), the contribution of each region becomes DCI. We show that our method dramatically simplifies the calculations. As a demonstration, we calculate the slightly off-shell DCI pentabox integral up to power corrections. The contributions of all 32 regions appear to be expressible in terms of products/ratios of $\Gamma$-functions multiplied by some powers of DCI cross-ratios. Therefore, after removing the regularization, we obtain the final expression in terms of cross-ratios logarithms only. We have checked that our result for pentabox integral numerically agrees with the result of the recent Belitsky\&Smirnov paper [arXiv:2508.14298] which has essentially more complicated form.