$k^2 \gg m^2$ and $k^2 \gg \mu^2$, it takes up the form ${\cal F} \sim k^2 \log \left( \frac{k^2}{\mu^2} \right)$.

It is important to notice that all this holds good even when we regularize the UV divergent integrals in some other way. Similarly, it is not just a property of this theory: it can be easily seen for other theories (e.g. Yukawa theory, $\phi^4$ theory, QED): in general, the pole part of any sub-divergence-free diagram is a polynomial in its external momenta. It has been argued that this holds good also in other theories, where the residues of the poles always contain the external momenta and masses as low-order polynomials (see [@'tHooft1972189], [@PhysRevD.10.1201], ch. 9 of [@kleinert2001critical], pg. 148 of [@hooft1994under]). Thus, a polynomial in external momenta plus a function which is logarithmic in external momenta is a generic feature of Green’s functions in usual QFT.

At this point it is worth re-emphasizing that if we wish to find the number of counter terms needed to renormalize a correlator or the form of these counter terms (things we wish to find, even for a non-renormalizable theory), the dependence of the Green’s function on external momenta is a very important tool.

