  The amplitude for multiple soft gluon emissions, ordered in their relative
softness, can be written as a product of infrared (IR) divergent loop-expanded
soft gluon currents and the matrix elements. In a more recent work,
Angeles-Martinez, Forshaw and Seymour \cite{Angeles-Martinez:2016dph} (AMFS)
showed that the result for this amplitude can in fact be re-expressed in an
ordered evolution approach, involving \textit{IR finite} one-loop insertions
where the virtual loop momentum is constrained in a highly non-trivial way by
the $k_T$ of the adjacent real emissions. The result thus exhibits a novel
amplitude level QCD coherence where the IR divergences originating only from
the very last, softest, gluon emission remain, and the rest cancel. The proof
of the AMFS result at one-loop in QCD, however, involves many diagrams, and
only after carefully grouping and summing over all the diagrams does the
correct ordering variable emerge, making the higher order extension a
challenging task. Moreover, the compact, Markovian nature of the final AMFS
result is suggestive of a deeper underlying physics that is obscured in the
derivation using traditional diagrammatic QCD. By considering a (recursive)
sequence of effective field theories (EFTs) with Glauber-SCET operators, we
present an elegant derivation of this result involving only a handful of
diagrams. The SCET derivation offers clean physical insights, and makes a
higher order extension of the AMFS result tractable. We also show that the
grouping of QCD graphs necessary to derive the AMFS result in full theory is
already implicit in the Feynman rules of Glauber-SCET operators such that the
same result can alternatively be derived with significantly less effort in a
single EFT with multiple ordered soft gluon emissions.
