All-order vanishing of shockwave recoil operators

Prove that all shockwave recoil operators vanish in the light-cone gauge at every Self-Force order, so that the heavy-particle trajectory variation remains a straight line order by order in the Self-Force expansion.

Background

The Self-Force Effective Field Theory contains recoil operators describing fluctuations of the primary, highly energetic particle around its unperturbed straight-line trajectory. In the light-cone gauge defined by $\xi^\alpha h_{\alpha\beta}=0$, the paper explicitly demonstrates that the recoil contributions vanish at 1SF and 2SF order because the relevant contractions with the graviton propagator contain the null Killing vector $\xi^\alpha$.

The authors extend this observed simplification to an all-order conjecture: at every Self-Force order, the recoil operators should vanish in this gauge, implying that the heavy-particle variation is a straight-line deflection. Establishing this would provide an all-order structural result for the Self-Force EFT on a gravitational shockwave background.