$G$ is another dimensionless function and when $\eta = 0$, it is of the form $$G = 1 + \delta \log (-c H \eta_k) + \cdots \; ,$$ where $c$ is an ${\cal O}(1)$ constant. This changes the logarithmic running to $\log (H/\mu)$ [@SZ2010].

Counterterms
------------

A careful look at the argument presented by Senatore and Zaldarriaga [@SZ2010] (to establish that the logarithmic running is $\log (H/\mu)$) also tells that when $\eta = 0$, only one kind of divergence is present. This may suggest from the arguments in familiar applications of QFT (Sec. \[sec:usual\_qft\]) that we need just one counterterm to cancel the UV divergences in the two-point function in this theory. Thus, when $\eta = 0$, we have (see [@SZ2010]), $$\tilde{\cal G}_{\rm 1-loop} \sim \frac{1}{k^3} 
 \left( \frac{F_{-1}}{\epsilon} + \frac{F_{-1}}{2} \log \left( \frac{H(\eta_k)}{\mu} \right) + F_0 + \cdots \right) \; ,$$ where a factor of $(c_3^2 M^8/\epsilon^4 M{\rm Pl}^8)$ is understood to sit in the front, apart from some numerical factors. This implies that $$\label{G_loop_asym}
 \tilde{\cal G}_{\rm 1-loop} \sim k^3 \tilde{\cal G}_{\rm tree}^2 
 \left( \frac{F_{-1}}{\epsilon} + \frac{F_{-1}}{2} \log \left( \frac{H(\eta_k)}{\mu} \right) + F_0 + \cdots \right) \; ,$$ with a factor of $(c_3^2 M^8/\epsilon^2 M_{\rm Pl}^4 H^4)$ in the front. 