Loop-level dipole currents and the renormalized hard celestial current algebra in QED

We determine the finite-energy action of the normalized one-loop logarithmic soft-photon operator in an infrared-subtracted abelian gauge theory.

Its commutator with Mellin-difference hard currents has a scheme-independent hard-hard residue that survives every one-particle redefinition.

With the meromorphic continuation stated explicitly below, a two-particle Plancherel transform identifies this residue with an analytic two-particle primary module, and the coefficient map is a hard-current one-cocycle.

The cocycle defines a minimal filtered abelian extension.

It has a canonical two-particle primitive and integrates to an affine action.

For scalar hard legs, the fixed-leg operator agrees coefficient by coefficient with the symmetry-governed long-range logarithmic tower of Choi, Kadhe, and Puhm.

Applied to a tree-level scalar-QED photon-exchange block, the construction determines the logarithmic two-particle coefficient functional from the ordinary hard amplitude and the universal soft kernel.

This gives a finite-energy relation between the dipole-current Ward identity and the exponentiated long-range celestial OPE.