the integrals which were encountered in the pure gravity calculation [@mr] already: $$\begin{aligned}
\Phi^p_n(w)&=\frac{1}{\Gamma(n)}\int_0^\infty{\textrm{d} z\:}z^{n-1}\frac{R^{(0)}(z)-z R^{(0)}{}'(z)}{[z+R^{(0)}(z)+w]^p}\qquad & (n&>0)\\
\Phi^p_0(w)&=(1+w)^{-p} & (n&=0)
\end{aligned}$$ The second contribution on the RHS of (\[4d1loop\]) is the familiar “asymptotic freedom” term due to the self-interaction of the gauge bosons, while the first one, due to the virtual gravitons, is new.

Several comments are in order here.

[**(1)**]{} The gravitational correction is manifestly cutoff scheme dependent, i.e. it depends, via $\Phi^1_1(0)$, on the shape function $R^{(0)}$. However, for any admissable choice of $R^{(0)}$ the constant $\Phi^1_1(0)$ is positive. As a result, the gravity term has a qualitatively similar impact on ${g_{\textrm{YM}}}(k)$ as the gauge boson loops, namely to drive ${g_{\textrm{YM}}}(k)$ smaller at larger $k$. It tends to speed up the approach of asymptotic freedom.

For the exponential cutoff $R^{(0)}(y)=y/(e^y-1)$, for instance, one finds $\Phi^1_1(0)=\pi^2/6$, while the “optimized” one [@opt], $R^{(0)}(y)=(1-y)\Theta(1-y)$, yields $\Phi^1_1(0)=1$.

[**(2)**]{} The gravitational correction, in perturbative language, originates from a quadratic divergence or, in FRGE language, a quadratic running with $k$. For this reason its scheme dependence is by no means surprising or alarming. Rather, it is the usual situation which is always encountered when the effective average action is applied to matter theories 