Non-linear evolution in QCD at high-energy beyond leading order

Abstract

The next-to-leading order (NLO) Balitsky-Kovchegov (BK) equation describing the high-energy evolution of the scattering between a dilute projectile and a dense target suffers from instabilities unless it is supplemented by a proper resummation of the radiative corrections enhanced by (anti-)collinear logarithms. Earlier studies have shown that if one expresses the evolution in terms of the rapidity of the dilute projectile, the dominant anti-collinear contributions can be resummed to all orders. However, in applications to physics, the results must be re-expressed in terms of the rapidity of the dense target. We show that although they lead to stable evolution equations, resummations expressed in the rapidity of the dilute projectile show a strong, unwanted, scheme dependence when their results are translated in terms of the target rapidity. Instead, in this paper, we work directly in the rapidity of the dense target where anti-collinear contributions are absent but where new, collinear, instabilities arise. These are milder since disfavoured by the typical BK evolution. We propose several prescriptions for resumming these new double logarithms and find only little scheme dependence. The resummed equations are non-local in rapidity and can be extended to full NLO accuracy.

HERA data and collinearly-improved BK dynamics

“…(6), actually goes beyond the original proposal from Ref. [45] by additionally including a class of DGLAP-like single transverse logarithms (both collinear and anti-collinear) which appears in the NLO BK equation. While in principle one can also include [45] the full set of NLO BK corrections in our evolution equation, this is numerically cumbersome (see however [42]) and it goes beyond the scope of this Letter.…”

“…At a first sight, these strategies seemed to be successful, leading to stable evolution equations [41,42] and allowing for good fits to the small-x HERA data [43,44]. However, a recent study [45] revealed some inconsistencies in the original analyses in [31,41,43,44]. In particular, there was a confusion concerning the meaning of the rapidity variable which plays the role of the evolution time.…”

“…2 for details). When translated to this physical variable, the results of the original resummations in [31,41] show a strong scheme dependence, preventing any meaningful phenomenological applications [45]. The success of the corresponding fits to the HERA data [43,44] was rather fortuitous and can be attributed to several factors: (i) these fits blindly assumed the physical rapidity variable η = ln(1/x) (although this was inconsistent with the resummation scheme), (ii) the rapidity interval over which one can probe high-energy evolution is relatively limited making it delicate to critically probe the effects of resummation.…”

“…To overcome these difficulties, Ref. [45] proposed a reorganisation of perturbation theory in which the evolution time is the physical rapidity η. This program lead to a new version of "collinearly-improved BK equation", shown below in Eq.…”

“…Moreover, they are partially suppressed by non-linear effects, so their resummation is somewhat less critical. As a consequence, a comparison between various resummation prescriptions shows only little scheme dependence [45], at the level of the expected perturbative accuracy of the resummed equation.…”