Conformal transformations of the S-matrix; $\beta$-function identifies change of spacetime

First conformal transformations of the $S$-matrix are derived in massless ${\varphi }^4$-theory. Then it is shown that the anomalous transformations can be rewritten as a symmetry once one has introduced a local coupling and interprets the charge of the symmetry accordingly. By introducing a suitable effective coupling on which the $S$-matrix depends one is able to identify via the $\beta$-function an underlying new spacetime with non-trivial conformal (flat) metric.