Lemma: Actual relative scaling degrees

In four relative coordinates,

The same upper bounds hold for $G_R$ and its first derivatives. Every nonzero numerical coefficient in (eq:Wickexpansion) therefore has degree below four.

Proof. Changing spatial momentum variables gives

The distributions $W_{\mu }$ converge to $W_0$ as $\mu \downarrow 0$. After pairing with a spacetime test, its spatial Fourier transform decreases rapidly in momentum, uniformly for time in a fixed compact interval. At small momentum,

which is locally integrable in three dimensions. Dominated convergence applies.

Splitting the time integration into positive and negative parts gives the same convergence for ${W_{\mu }}^{\mathrm{F}}$. Positive dilations preserve those two time regions, so

The nonzero massless limits give degree two; differentiation adds at most one inverse power of $\lambda$. Equation (eq:Widentities) gives the retarded bounds. The mass $m$ is fixed in this scaling operation.