Longitudinal rescaling and high-energy effective actions

Abstract

Under a longitudinal rescaling of coordinates x 0,3 → λ x 0,3 , 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 sharp-momentum cut-off gauge invariant. Our result for the quantum action is found, assuming | ln λ | ≪ 1, which is essential for the validity of perturbation theory. If λ is sufficiently small (so that | ln λ | ≫ 1), then the perturbative renormalization group breaks down. This is due to uncontrollable fluctuations of the longitudinal chromomagnetic field.

Renormalization of gauge theories on general anisotropic lattices and high-energy scattering in QCD

“…The evaluation of the various terms is performed generalising the techniques developed in [36] to the anisotropic case. Since such a generalisation is straightforward, here we simply list the 10 In these quantities one should in principle include also the higher-order terms mentioned above in footnote 9, by properly redefining fµν . 11 This clearly remains true also if higher-order terms neglected in Eq.…”

“…The results of Section 2 can be used to obtain some insight in the approach to high-energy scattering based on longitudinally rescaled actions [6,7,8,9,10,11,12]. The physical idea behind this approach is that in high-energy scattering processes the longitudinal directions appear highly Lorentz-contracted, so that it should be possible to achieve an effective description through an appropriately rescaled action.…”

“…Anisotropic lattices provide, quite obviously, the natural setting for the nonperturbative study of anisotropic systems, also beyond numerical applications. An interesting case is that of longitudinally rescaled actions, which in recent years have been considered in the context of highenergy scattering in QCD [6,7,8,9,10,11,12]. The basic idea of Refs.…”

“…[6,7,8,9] only the classically rescaled action was considered, while the important effect of quantum corrections was studied later in Refs. [10,11,12], in the framework of Wilsonian anisotropic renormalisation in the continuum. In this context, the use of a gauge-invariant, anisotropic lattice regularisation could lead to more insight in the structure of quantum corrections.…”

“…In Section 4 we discuss the longitudinally rescaled actions of Refs. [6,7,8,9,10,11,12], focussing on the representation of the SU (N c ) gauge theory as a set of coupled two-dimensional principal chiral models. Finally, Section 5 contains our conclusions and prospects.…”