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

Abstract

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.