Ordering multiple soft gluon emissions using SCET with Glauber operators

In QCD, the amplitude for multiple soft gluon emissions has been shown to exhibit a form of coherence, wherein one-loop corrections are rendered IR finite by the transverse momentum of the real emissions. Using a sequence of soft-collinear effective theories (SCET) with Glauber operators, we present a compact and elegant derivation of this property. Our derivation highlights an important role played by the Lipatov vertex. The EFT approach provides increased physical intuition and, for the first time, a clear road map for how this result can be extended to higher loop orders.

Figure 1

Relation of limits of the virtual loop insertions to the adjacent dipole emission momenta in the ordered soft gluon emission result. The graphs (a), (b) and (c) correspond to the three lines in Eq. (7).

Figure 2

Diagrams needed to reproduce the imaginary part of one-loop and up to two real soft gluon emission amplitude. Labels $i$, $j$ correspond to hard particles. The gluon colored in blue is more energetic of the two ($q_1$). The green solid lines correspond to off-shell propagators. They have been depicted in this way to make the color ordering apparent. Graphs with Glauber exchange between a soft gluon and its parent are zero in Feynman gauge, and not shown. (a)The Glauber exchange graph in ${\mathrm{EFT}}_n$, (b) Single soft gluon emission graphs with a Glauber exchange in ${\mathrm{EFT}}_n$. The second graph involves the Lipatov vertex. Additional graphs with permutations of $i$ and $j$ are not shown, (c) Glauber exchange graphs in ${\mathrm{EFT}}_{n+1}$, and (d) An indicative selection of single soft gluon emission graphs with Glauber exchanges in ${\mathrm{EFT}}_{n+1}$. One must also consider other allowed permutations of gluon 2 and the Glauber vertices.