  Conformally soft operators and their associated soft theorems on the
celestial sphere encode the low energy behaviour of bulk scattering amplitudes.
They lead to an infinite dimensional symmetry algebra of the celestial CFT at
tree-level. In this paper, we introduce new operators in the celestial CFT in
order to extend the definition of conformally soft currents to include one-loop
effects. We then compute their OPEs with other operators in the theory. We also
examine new subtleties that arise in defining OPEs of two conformally soft
operators. We elucidate the connection between the new operators and loop
corrected soft theorems in the bulk. Finally, we conclude by demonstrating how
these operators fit into the framework of a logarithmic CFT.
