A particle’s position is ill-defined at the wavepacket scale. Another approach would be to reverse-engineer $b$ from the deflection angle, which is the approach of Eq. (2).

The vector $A^{\mu }$ is manifestly invariant under the shift $S^{\mu }\to S^{\mu }+\xi q^{\mu }/q^2$ of Ref. [50] where $q^{\mu }=-k_2^{\mu }-k_3^{\mu }$. The invariance of the amplitude under the shift is violated because the terms dependent on $p_1·S=0$ are dropped when products of Levi-Civita tensors are reduced, while $p_1·q\ne 0$. The exact shift invariance would be restored if all $p_1·S$ were kept. Further discussions can be found in updated versions of Ref. [29].

While the canonical algebra satisfied by $S^{\mu \nu }$ inherited from $J^{\mu \nu }$ is tied to a different spin supplementary condition [74], the covariant one is chosen because the little group space is defined in the particle’s rest frame. Identifying spin operators as the rotation generators in the rest frame of the particle is equivalent to choosing the covariant spin supplementary condition.

The spinor notations follow that of Ref. [40] where incoming positive helicity states are associated with $|p]$ spinors. Normalization for the Levi-Civita tensor is ${\epsilon }^{0123}=+1$.

The use of four dimensional helicity scheme [78, 79] in dimensional regularisation of the loop integral is implicit, since four dimensional spinor variables were used to construct the integrand.

Since helicity nonpreserving amplitudes are present at tree level for graviton scattering, the eikonal should rather be treated as a matrix with ${\delta }_{\pm }$ as diagonal entries and remainder terms should be calculated using matrix algebra [88, 89]. Separate treatment of ${\delta }_{+}$ and ${\delta }_{-}$ is justified as helicity nonpreserving off-diagonal terms are suppressed at the considered order in $(\omega b{)}^{-1}$ (29).

As a curiosity, it is remarked that the quantum terms can be absorbed by redefining $m\to m^{\prime }=m+\omega$, canceling the quantum remainder terms of the spinless sector at the same time. This may have connections to the difference between the on-shell mass and the classical mass [66]. However, the shift fails to cancel the quantum remainder terms of spin-linear sector.