Longitudinal Rescaling and High-Energy Effective Actions

Under a longitudinal rescaling of coordinates x0,3 → λx0,3, λ ≪ 1, the classical QCD action simplifies dramatically. This is the high-energy limit, as λ ∼ s−1/2, where s is the center-of-mass energy squared of a hadronic collision. We find the quantum corrections to the rescaled action at one loop, in particular finding the anomalous powers of λ in this action, with λ < 1. The method is an integration over high-momentum components of the gauge field. This is a Wilsonian renormalization procedure, and counterterms are needed to make the sharpmomentum cut-off gauge invariant. Our result for the quantum action is found assuming | lnλ | ≪ 1, which is essential for the validity of perturbation theory. If λ ≪ 1, then the quantum action becomes strongly coupled. It is therefore impossible to derive this action from either classical considerations or perturbation theory. We briefly discuss the implications for effective theories of high-energy collisions.