Longitudinal Rescaling and High-Energy Effective Actions

Under a rescaling of longitudinal coordinates $x^{0,3}$ by a factor $\lambda$ which is taken to zero, the classical QCD action simplifies dramatically. This is the high-energy limit, as $\lambda$ is of order $s^{-1/2}$, 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 $\lambda$ in this action, for $\lambda$ close to unity. 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 sharp-momentum cut-off gauge invariant. Our result for the quantum action is found, assuming that the logarithm of $\lambda$ is small, which is essential for the validity of perturbation theory. If $\lambda$ is sufficiently small (so that its logarithm is large), then the perturbative renormalization group breaks down. This is due to uncontrollable fluctuations of the longitudinal chromomagnetic field.