Locality of renormalized actions in momentum space

Question

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?

Comments

The methods used by Dütsch are tailored to x-space; in place of  differential renormalization one can use analytic regularization; see Section 3.5 of the book.

• M. Dütsch, From Classical Field Theory to Perturbative Quantum Field Theory, Birkhäuser 2019.

I don't think that these techniques are more difficult to apply than momentum space methods. Also, the former (unlike the latter) are applicable in curved backgrounds.

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@ArnoldNeumaier I forgot to mention that that is the book where I learned about this. From what I can tell it looks like something which gets easier with practice, but I'm currently familiar with regularization in p-space. Do you know if there is a paper on applying differential renormalization to experiment e.g. calculating cross sections? Or is the procedure of renormalization most easily done in different forms for different goals (for flat space)? I think if I see it in different contexts I would be more comfortable using it.

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The two methods are compared in

D. Prange, Epstein-Glaser renormalization and differential renormalization. Journal of Physics A: Mathematical and General, 32 (1999), 2225.

which might help you.

And there was a typo in my previous comment: in place of  differential renormalization one can use analytic regularization!