The breaking of homogeneous scaling by distributions appearing in a QFT may be used to calculate elements of the Gell-Mann Low cocycle. To $n$th order in perturbation theory, $$Z_n=\sigma_\rho\circ T_n\circ \sigma_\rho^{-1} - T_n$$.

From which the renormalized action can be obtained. M. Dütsch elaborates on this in x-space using differential renormalization: The resulting scaling difference is local i.e. depends only on (convolutions with) delta functions and their derivatives.

Since differential renormalization is not, at least in introductory texts, a popular method, I was wondering if the results can be straightforwardly written in momentum space with e.g. Epstein-Glaser renormalization, but it is unclear how the scaling difference maintains locality despite supposedly dealing with the same expressions. In p-space, the scaling law takes the form (see G. Scharf) $$\sigma_\rho\circ t\circ \sigma_\rho^{-1}=\rho^\omega \hat t\left(\frac{k}{\rho}\right)$$

but when applied to common loop distributions such as the scalar vacuum polarization distribution (eq 230 of https://arxiv.org/pdf/0906.1952.pdf) I do not see any cancellation of terms nonpolynomial in $k$, which to my knowledge corresponds to nonlocal terms in x-space.

**Can the elements of the cocycle and the renormalized action be calculated in p-space (and if so, how), or is this so far only understood for x-space calculations?**