L.6 Renormalization scale

The extension of L.4 depends on the fixed cutoff function $\varpi \in {C_c}^{\infty }({\mathbb{R}}^4)$ with $\varpi (0)=1$. This section varies that prescription by dilating $\varpi$, producing a one-parameter family of coherent extensions. The resulting $\mu$-dependence is a change of subtraction prescription; it is not identified with a physical renormalization-group flow of a complete gravitational quantity (Remark rem:Lscale).

scale and dilated cutoffs. Fix a reference ${\mu }_0>0$. For $\mu >0$ define

Then ${\varpi }_{\mu }\in {C_c}^{\infty }({\mathbb{R}}^4)$ and ${\varpi }_{\mu }(0)=\varpi (0)=1$. At $\mu ={\mu }_0$ one recovers ${\varpi }_{{\mu }_0}=\varpi$.

projections. For $\rho \in \{0,1,2,3,4\}$ form the matrices and functions of (eq:L-Wrho) with $\varpi$ replaced by ${\varpi }_{\mu }$:

Each $M^{(\rho ,\mu )}$ remains unitriangular (Leibniz; ${\varpi }_{\mu }(0)=1$), so invertible, and ${W^{(\mu )}}_{\rho }$ projects onto the test functions vanishing to order $\rho$ at $0$. At $\mu ={\mu }_0$ one has ${W^{({\mu }_0)}}_{\rho }=W_{\rho }$ of (eq:L-Wrho).