Celestial Current Algebra from Low's Subleading Soft Theorem

The leading soft photon theorem implies that four-dimensional scattering amplitudes are controlled by a two-dimensional (2D) $U(1)$ Kac-Moody symmetry that acts on the celestial sphere at null infinity ($\mathcal{I}$). This celestial $U(1)$ current is realized by components of the electromagnetic vector potential on the boundaries of $\mathcal{I}$. Here, we develop a parallel story for Low's subleading soft photon theorem. It gives rise to a second celestial current, which is realized by vector potential components that are subleading in the large radius expansion about the boundaries of $\mathcal{I}$. The subleading soft photon theorem is reexpressed as a celestial Ward identity for this second current, which involves novel shifts by one unit in the conformal dimension of charged operators.

to hold in a more general context, as their form is largely dictated by symmetries.
This note is organized as follows. In Section 2, we introduce our conventions and present basic formulas. In Section 3, we rewrite the conservation law as a relation between the subleading soft currents and the hard charged currents. In Section 4, we take the quantum matrix element of this conservation law and express it as a Ward identity for a novel 2D current algebra on the celestial sphere. Appendix A gives some details of the asymptotic expansion about I in Lorenz gauge.

Maxwell Equations in Lorenz Gauge
We largely employ the retarded (advanced) coordinates on flat Minkowski space ds 2 = −du 2 − 2dudr + 2r 2 γ zz dzdz = −dv 2 + 2dvdr + 2r 2 γ zz dzdz, with u (v) retarded (advanced) time and γ zz = 2/(1 + zz) 2 the unit round metric on S 2 . These are related to the Cartesian coordinates (x 0 , x 1 , x 2 , x 3 ) by In this paper we use the the Lorenz gauge condition ∇ µ A µ = 0. The Maxwell equations ∇ µ F µν = e 2 j ν in this gauge in retarded coordinates are See Appendix A for further details.
3 Conservation Law on I + Low's subleading soft photon theorem was recently shown to be the quantum matrix element of a charge conservation law [13]. The conserved charge on I + is: where the charge is parameterized by a real vector field Y z on I + − , the past boundary of I + . 1 The fields in this expression are the functions of (u, z,z) that appear as coefficients in the asymptotic 1 r expansion about I + . The order 1 r n at which they appear in this expansion is denoted by the superscript (n). For simplicity, we restrict here to the case where there are no long range magnetic fields near spatial infinity so that A (0) z is pure gauge and F (0) For the leading soft charge, the analog of the first 'soft' term in Q + is a total u-derivative and reduces to a difference between two terms on the boundaries of I + , signalling the central role of I boundary dynamics. In contrast, this total derivative structure is not manifest in the soft term in (4). However, we now show that this structure reappears when Q + is reexpressed in terms of the subleading component A Inverting the action of Dz on A (0) z , integrating over u and assuming the hard currents vanish at the boundaries I + ± gives Lorenz gauge ∇ µ A µ = 0 leaves unfixed residual gauge transformations of the form A µ → A µ + ∂ µ ε with ε = 0. The solution to this equation in retarded coordinates requires two pieces of free data, at different orders in the asymptotic expansion: the free function ε (0) (z,z), which is related to the leading soft theorem, and the free function ε (1) (u, z,z), which is independent free data. This latter residual freedom enables us to fix the subsidiary gauge condition which implies that ∂ u ε (1) = 0. We are left with a free function ε (1) (z,z). The gauge transformations are parametrized as At early and late times along future null infinity, where the matter current is zero, the field configurations return to pure gauge. Hence the asymptotic behavior near I + ± is where the tilde denotes a log r dependence (see Appendix A for details) and where the boundary fields ϕ shift under gauge transformations as ϕ ± +ε (0) and ϕ (1) ± +ε (1) . The difference in their values at I + + and I + − is determined by the hard charges and cannot be gauge-fixed to zero. To underscore this, we rewrite (6) as Similarly, on I − , To write the conservation law as a shift, we need to look at both I + and I − . Charge conservation is the identity Q + = Q − . Setting Yz = 1 2π(z−z ′ ) so that D z Yz is a delta function, and using the antipodal matching ϕ (1) the conservation law is (13) Defining the subleading soft photon current (13) can finally be written 2 We see that, if there is any charge flux j z and j u , it is impossible to set A

Celestial Current Ward Identity
In a quantum setting, the conservation law (15) becomes operator identities whose S-matrix elements are equivalent to the subleading soft photon theorem. We denote this z n , . . . |Q + S − SQ − |z 1 , . . . = 0, where we consider a state |z 1 , . . . , with n massless hard scalar particles of energies ω k , charges eQ k and momenta p µ k . The total charge current for massless scalars Φ k with charge Q k is The canonical commutator of the current component j (2) w with the leading term in Φ k at I + is where D w ≡ γ ww ∂w. In terms of the Fourier transform Φ (1) the action of the hard currents in the charge (10) becomes 3,4 It is illuminating to rewrite the scattering amplitudes as correlation functions on the celestial sphere, adopting the compact notation [7] z n+1 , . . .
The subleading soft theorem then becomes The Mellin transform to a conformal basis for particles with helicity s with conformal weights is simply In this conformal basis, (22) becomes the current algebra relation This is the celestial representation of the subleading soft theorem. 3 Up to irrelevant contact terms which will vanish after contour integration. 4 Using −iωΦ (2) kω = γ zz DzDzΦ (1) kω , the right hand side can be rewritten as . This suggests a connection with the identification in [14] of subleading soft symmetries with gauge transformations that diverge linearly with r. It would be interesting to understand this better.
The operators O which create spacetime particles in a conformal basis appearing in celestial amplitudes are in different types of representations -typically the continuous unitary principal series -than those we are accustomed to in standard 2D CFT. The corresponding amplitudes take a rather different form often involving delta functions on the sphere [16,17,18,19], which makes possible relations between amplitudes with shifted conformal weights. Relations of this general type were noted in the gravitational context in [20] and verified by Stieberger and Taylor [21] in some special cases. It would be of interest to examine (25) in explicit examples.
Finally, we note that integrating around a contour C weighted by a holomorphic function ε(z), the subleading soft theorem takes the alternate form where the sum is restricted to operators inside the contour.
In order to consistently solve the Maxwell equations in ∇ µ A µ = 0 gauge we must allow logarithmic falloffs in the gauge fields. This gives the expansion