$H/M_{\rm Pl}$ [@2013arXiv1303.5082P]. If we knew the actual values of these parameters (rather than just the limits), we expect that we could work in the On-shell (OS) renormalization scheme to perform the actual renormalization of the theory. In usual QFT, while renormalizing the two-point function in OS scheme, we choose the counter-terms such that (1) the divergent part in the loop integral (the $1/\epsilon$ term) gets canceled, (2) the dependence on the fake renormalization scale gets cancelled, and, (3) the rest of the part of the counterterm is chosen such that $\Pi(-m^2) = 0$ and $\Pi'(-m^2) = 0$, see [@quantum], these conditions ensure that the parameter $m$ appearing in the Lagrangian is the physical mass and the field strength is normalized (thus, at $k^2=-m^2$, the propagator has a simple pole with unit residue). In Eq. (\[G\_loop\_asym\]), we can use $$\log \left( \frac{H(\eta_k)}{\mu}\right) = 
 \log \left( \frac{H(\eta_k)}{H(\eta_{k_0})}\right) + \log \left( \frac{H(\eta_{k_0})}{\mu}\right) \; ,$$ where $k_0$ is a pivot scale to absorb the $\mu$ dependent term in the CT, but instead of the two conditions (given by Lehmann-Källén form of the exact propagator), in the usual flat spacetime field theory, we shall need three (since we have three CTs) and we have 