  In this paper, we study loop corrections to the recently proposed new soft
theorem of Cachazo-Strominger, for both gravity and gauge theory amplitudes. We
first review the proof of its tree-level validity based on BCFW recursion
relations, which also establishes an infinite series of universals soft
functions for MHV amplitudes, and a generalization to supersymmetric cases. For
loop corrections, we focus on infrared finite, rational amplitudes at one loop,
and apply recursion relations with boundary or double-pole contributions. For
all-plus amplitudes, we prove that the subleading soft-theorems are exact to
all multiplicities for both gauge and gravity amplitudes. For single-minus
amplitudes, while the subleading soft-theorems are again exact for the
minus-helicity soft leg, for plus-helicity loop corrections are required. Using
recursion relations, we identify the source of such mismatch as stemming from
the special contribution containing double poles, and obtain the
all-multiplicity one-loop corrections to the subleading soft behavior in
Yang-Mills theory. We also comment on the derivation of soft theorems using
BCFW recursion in arbitrary dimensions.
