  At leading order, the $S$-matrices in QED and gravity are known to factorise,
providing unambiguous determinations of the parts divergent due to infrared
contributions. The soft $S$-matrices defined in this fashion are shown to be
defined entirely in terms of $2$ dimensional models on the celestial sphere,
involving two real scalar fields, allowing us to express the soft $S$-matrices
for real as well as virtual divergences as two dimensional correlation
functions. We discuss what this means for finding holographic representations
of scattering amplitudes in QED and gravity and comment on simple double copy
structures that arise during the analysis.
