Renormalized position space amplitudes in a massless QFT

Ultraviolet renormalization of massless Feynman amplitudes has been shown to yield associate homogeneous distributions. Their degree coincides with the degree of divergence while their order - the highest power of the logarithm in the dilation anomaly - is given by the number of (sub)divergences. We observe that (convergent) integration over internal vertices does not alter the total degree of (superficial) ultraviolet divergence. For a conformal invariant theory internal integration is also proven to preserve the order of associate homogeneity. Our conclusions concerning the (off-shell) infrared finiteness of the ultraviolet renormalized massless $\varphi^4$ theory agrees with the old result of Lowenstein and Zimmermann [LZ]