See [2, 3] and references within.

For an alternative approach, the so-called corner symmetry construction, see, e.g., [17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

We thank Hyungrok Kim for pointing out that the operator we define in (18) is related to the Todd polynomial.

Generally we will take $\mathfrak{r}$ to be the radial direction, but we will allow for other possibilities as well (see, e.g., [28] for an example in light-cone gauge, where $\mathfrak{r}=V$).

This could arise as a result of imposing further constraints, see, e.g., [15], where the field-dependent terms are a consequence of imposing Lorenz gauge.

For other works where Goldstone modes appear in the context of soft limits and celestial holography, see, e.g., [47, 48, 49, 50, 51].

This assumption is supported by the polynomial tree level falloffs in the $u$ coordinate which are directly related to the falloffs in $r$ in radial gauge, see Sec. 3 of [39]. Other gauges, such as harmonic gauge, cannot be consistent for a $1/r$ expansion, e.g., [15].

Here we use the convention about symmetrization and antisymmetrization of indices where $T_{(\mu \nu )\alpha \dots }$ and $T_{[\mu \nu ]\alpha \dots }$ are defined as $\frac{1}{2}(T_{\mu \nu \alpha \dots }+T_{\nu \mu \alpha \dots })$ and $\frac{1}{2}(T_{\mu \nu \alpha \dots }-T_{\nu \mu \alpha \dots })$, respectively, for some tensor $T$.

For some previous work on double copy relations at null infinity, see also [65, 66, 67, 68, 69].

For recent results at the first subleading level, see [34].