Off-Shell CHY Amplitudes

TL;DR

This work extends CHY scattering formulas to off-shell kinematics by promoting the scattering functions to off-shell versions $\hat{f}_i$ built from $\hat{s}_{ij}=s_{ij}+x_{ij}$, with $x_{ij}=x_{ji}$ chosen to preserve Möbius invariance and reproduce the correct propagator structure $s_{i_1\cdots i_p}$ of field theory. By explicit analysis of $n=4$ and $n=5$, the authors derive concrete $x_{ij}$ shifts for massless and massive scalars, and demonstrate how the off-shell amplitudes retain the same pole structure as their on-shell counterparts; they also extend the construction to a gauge amplitude, yielding a Möbius-invariant off-shell extension. The results reveal a color-dependent feature in off-shell scatterings and provide a general framework for arbitrary $n$ and color configurations, with a clear path to incorporating masses via $m^2$ shifts. Overall, the paper establishes a coherent OFF-SHELL CHY program, linking off-shell Green’s functions to the CHY formalism and highlighting avenues for further study of gauge theories in this off-shell CHY setting.

Abstract

The Cachazo-He-Yuan (CHY) formula for on-shell scattering amplitudes are extended off-shell. The off-shell amplitudes are Möbius invariant, and have the same momentum poles as the on-shell amplitudes. The same technique is also used to obtain off-shell massive scalar and vector boson amplitudes.