The Space-Cone Gauge, Lorentz Invariance and On-Shell Recursion for One-Loop Yang-Mills amplitudes

TL;DR

The paper develops an on-shell recursion framework for one-loop rational gluon amplitudes in the space-cone gauge, exploiting complex momentum shifts that preserve vertex structure and ensuring vanishing boundary behavior. It shows how Lorentz invariance and reference-twistor independence fix elusive soft factors that accompany double-pole contributions, illustrating with all-plus and (-++...) amplitudes up to six points and deriving a general pattern for arbitrary n. The results provide a gauge-driven, analyticity-based route to compute rational one-loop Yang-Mills amplitudes efficiently, connecting to and validating the Bern–Dixon–Kosower soft factors. Overall, the work highlights a deep interplay between gauge choice, Lorentz invariance, and on-shell recursion in perturbative QCD.

Abstract

Recursion relations are succinctly obtained for $(++... +)$ and $(-++... +)$ amplitudes in the context of the space-cone gauge in QCD. We rely on the helicity symmetry of the problems to dictate our choices of reference twistors and the momentum shifts to complexify the amplitudes. Of great importance is the power of gauge Lorentz invariance, which is enough to determine the soft factors in the latter cases.

Figures (6)

• Figure 1: $A_{12}$ and $A_{15}$
• Figure 2: $A_{45}^b$ and $A_{23}^a$ on the left side figure, and $A_{45}^a$ and $A_{23}^b$ to the right
• Figure 3: A group of cut sub-amplitudes, $A_{12}$ and $A_{16}$ which involve a five-point $(+++++)$ factor. Their sum is reference-twistor independent and given in (\ref{['12+16']}).
• Figure 4: These cut sub-amplitudes contain a lone-loop (++++) on-shell factor. Their sum is reference-twistor invariant (separately for the left and right cut sub-amplitudes), corresponding to (\ref{['6pta']}) and (\ref{['6ptb']}).
• Figure 5: A group of cut sub-amplitudes which adds up to a reference-twistor invariant expression, (\ref{['6pt14']}).