The dual conformal box integral in Minkowski space

The dual conformal box integral in Minkowski space is not fully determined by the invariants $z$ and $\bar{z}$. Depending on kinematic region its value a 'branch' of Bloch-Wigner function which occurs Euclidean case. Dual special transformations can change such way that jumps to another branch this function, encoding subtle breaking invariance for integral. We classify conformally equivalent configurations four points compactified space. show starting with any configuration, one reach up branches using transformations. also most real $\bar{z}$ be mapped configuration same two at infinity, where calculated directly only residue theorem.