Up to a subtlety recently pointed out in [14], arising from a special class of three-loop diagrams in the presence of massless charged particles.

The Celestial OPE of gluons and gravitons at one-loop were studied in [26, 27].

As is well known, in four spacetime dimensions both ${\mathcal{A}}_{N+1}^{(\mathrm{em})}$ and ${\mathcal{A}}_N^{(\mathrm{em})}$ are infrared (IR) divergent. However, since the soft factor appears as a ratio of these amplitudes, it is IR finite in the sense that the soft theorem in (1) relates the IR-finite hard parts of the two amplitudes, following the arguments of [16, 17, 36].

For the all-loop resumed $S$-matrix in a generic theory of QED, ${\mathcal{R}}_{N+1}^{(\mathrm{em})}$ cannot be related to ${\mathcal{A}}_N^{(\mathrm{em})}$, and hence it does not factorize. However, if we restrict to the tree-level $S$-matrix, then ${\mathbb{R}}_{\mathrm{em},N+1}^{(0,0)}$ factorizes and yields the well-known Low’s soft photon theorem, with a specific contribution from the nonminimal coupling through the field strength [38]. At one-loop order, ${\mathbb{R}}_{\mathrm{em},N+1}^{(0,0)}$ acquires nonuniversal contributions originating from different regions of the loop diagrams which fails to factorize [17].

In QED the high energy limit is not really well defined because of the existence of the Landau pole. We assume that the energies of the hard particles are well below that but they are much greater than the typical mass of a charged particle so that we can neglect the logarithmic and power law suppressed terms in (5) to a good approximation. However, it should be kept in mind that there is no region of parameter space where the correction terms are zero. It will be very interesting to study the soft symmetry interpretation of the subleading terms in the high energy expansion. We leave it for future investigation.

From (14) it is evident that the charge $Q_{\overline{d}}(\overline{w})$ defined through $Q_{\overline{d}}(\overline{w})=\oint \frac{d\overline{z}}{2\pi i}\overline{d}(\overline{w},\overline{z})$ measures the (anti-holomorphic) dipole moment of a charged operator, i.e., $$[Q_{\overline{d}}(\overline{w}),{\varphi }_{\mathrm{\Delta }}^{(\eta )}(z,\overline{z},e)]=-e(\overline{z}-\overline{w}){\varphi }_{\mathrm{\Delta }}^{(\eta )}(z,\overline{z},e),$$justifying the name “dipole current operator.”

In the context of Celestial holography composite operators have been studied in the recent literatures [44, 45].

Two component $SL(2,\mathbb{R}{)}_R$ spinor notation is given by $${\overline{Z}}^{\alpha }=\left(\begin{array}{c}\overline{z}\\ 1\end{array}\right),{\epsilon }_{\alpha \beta }=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),{\overline{Z}}_{\alpha }={\epsilon }_{\alpha \beta }{\overline{Z}}^{\beta }=\left(\begin{array}{c}1\\ -\overline{z}\end{array}\right),\alpha ,\beta =\pm 1/2.$$

In terms of the original field $\overline{d}(\overline{w},\overline{z})$ this is nothing but $⟨\overline{d}({\overline{w}}_1,{\overline{z}}_1)\overline{d}({\overline{w}}_2,{\overline{z}}_2)⟩=k\frac{{\overline{w}}_1-{\overline{w}}_2}{{\overline{z}}_1-{\overline{z}}_2}$. This form of the two point function is uniquely determined by the $SL(2,\mathbb{R}{)}_{\mathrm{R}}$ transformation law (18).

As an example the dimension $\overline{h}=1$ and spin-1 current ${\overline{d}}_{\alpha \beta }^1$ is given by $$\begin{gathered} {\overline{d}}_{\alpha \beta }^1(\overline{z})={\overline{d}}_{\beta \alpha }^1(\overline{z})=\underset{{\overline{z}}^{\prime }\to \overline{z}}{\lim }({\overline{d}}_{\alpha }^{\frac{1}{2}}({\overline{z}}^{\prime }){\overline{d}}_{\beta }^{\frac{1}{2}}(\overline{z})-k\frac{{\epsilon }_{\alpha \beta }}{{\overline{z}}^{\prime }-\overline{z}}) \\ ≕{\overline{d}}_{\alpha }^{\frac{1}{2}}(\overline{z}){\overline{d}}_{\beta }^{\frac{1}{2}}(\overline{z}):\mathrm{.} \end{gathered}$$

See also Refs. [33, 34] for closely related work.