with a quadratic running of parameters, masses, say. However, one should note that the couplings in $\Gamma_k$ as such are not observable or “physical” quantities. The latter must be $R^{(0)}$-independent. This independence comes about by a compensation of the scheme dependence among different running couplings. (In truncations this compensation might not be perfect.) In general there will be compensations between effective propagators and vertices, for instance. Analogous remarks apply to the gauge fixing dependence.

[**(3)**]{} The beta function for ${g_{\textrm{YM}}}^2$ depends on all three couplings, ${g_{\textrm{YM}}}^2$, $g$ and $\lambda$. In the approximation of (\[4d1loop\]) it happens to be independent of $\lambda$, but it does depend on $g(k)\equiv k^2 G(k)$, the dimensionless Newton constant. Hence the differential equation for ${g_{\textrm{YM}}}$ cannot be solved in isolation. In principle the full system (\[SDE\]) should be considered, and this would include the backreaction of the matter fields on the running of the gravitational parameters $g$ and $\lambda$. We shall not study this backreaction in the present paper. Instead, let us assume that the complete RG trajectory $k\mapsto({g_{\textrm{YM}}}(k),g(k),\lambda(k))$ admits a classical regime [@h3] in which Newton’s constant does not run appreciably so that we may approximate $$G(k)\approx G_0=\textrm{const},\quad g(k)=G_0 k^2\label{Gassumption}$$ This approximation, implicitly, has been 