Associativity of One-Loop Corrections to the Celestial Operator Product Expansion

There has been recent interest in the question of whether QCD collinear singularities can be viewed as the operator product expansion of a two-dimensional conformal field theory. We analyze a version of this question for the self-dual limit of pure gauge theory (incorporating states of both helicities). We show that the known one-loop collinear singularities do not form an associative chiral algebra. The failure of associativity can be traced to a novel gauge anomaly on twistor space. We find that associativity can be restored for certain gauge groups if we introduce an unusual axion, which cancels the twistor space anomaly by a Green-Schwarz mechanism. Alternatively, associativity can be restored for some gauge groups with carefully chosen matter.