Remarks on high energy evolution

Abstract

We make several remarks on the B-JIMWLK hierarchy. First, we present a simple and instructive derivation of this equation by considering an arbitrary projectile wave function with small number of valence gluons. We also generalize the equation by including corrections which incorporate effects of high density in the projectile wave function. Second, we systematically derive the dipole model approximation to the hierarchy. We show that in the dipole approximation the hierarchy has a simplifying property that allows its solution by solving the classical equation followed by averaging over the ensemble of initial conditions. I. INTRODUCTION.Recently there has been a lot of interest in the Balitsky-JIMWLK high energy evolution [1,2,3]. On one hand a lot of analytical and numerical work has been done to understand the mean field version of the hierarchy due to Kovchegov [4] -see for example [5,6]. On the other hand recently first steps has been taken to extend this hierarchy in order to include corrections away from the large density limit which are thought to be important for generating Pomeron loops [7], [8].Still some analytical aspects of the hierarchy itself are not well understood. For example, the original derivation due to Balitsky is for a probe containing a fixed small number of partons. A general derivation for a small probe with an arbitrary wave function has not been given. Such a derivation would be useful not only to polish the surface, but also to help understand how to go beyond the small probe approximation. Another aspect that has been discussed in the literature [9, 10] but not in a completely systematic way is how to get from the general JIMWLK equation to the dipole model [11], or large N c limit. In particular it is interesting to see whether the dipole limit allows one to probe numerically some of the fluctuations which are not included in the mean field approximation of [4], in a reasonably accessible way. This paper contains two sketches which deal with these issues. In Section 2 we present the derivation of the JIMWLK equation for a general projectile wave function. We go beyond the standard JIMWLK expression, by including terms which arise when the gluon density in the projectile is not small. Our derivation is purely algebraic and very simple. We hope this approach will be useful for casting the whole problem in a simpler framework and thus facilitate future progress. In Section 3 we discuss the dipole model limit of the evolution. We derive explicitly the dipole form of the evolution and also the 1/N c corrections to it, starting directly from the JIMWLK equation. We make an observation that the structure of the dipole evolution is such that it allows the solution of the whole system by solving Kovchegov's equation for a single dipole S-matrix with subsequent averaging over the ensemble of initial conditions. We suggest some initial distributions which directly probe fluctuations in the target wave function.

High energy asymptotics of scattering processes in QCD

“…Moreover, it helped to understand how to set up a systematics to go beyond the BK equation. It was recognized that the B-JIMWLK formalism was not complete [26,27], thus confirming what was expected [7,28,29]. The form of the new terms can be written down taking advantage of the correspondence to statistical physics.…”

NLO JIMWLK evolution unabridged

“…In addition, the Hamiltonian is invariant under the Z 2 transformation S → S † ; J L → −J R , which in [37] was identified as signature, and the charge conjugation symmetry S → S * . The JIMWLK Hamiltonian is derivable from perturbatively computable hadronic wavefunction [38][39][40]. At LO, the wavefunction schematically (omitting transverse coordinates and color indices) has the form…”

“…The algorithm of obtaining the Hamiltonian for high energy evolution starting from the soft gluon wave function has been described in detail in [38][39][40]. Given the general form eq.…”

The Yin and Yang of high energy chromodynamics: Scattering in black and white

“…The scattering matrix operator when acting on Ψ multiplies the gluon wave function by the matrix S [26]. We thus have…”

“…Although the JIMWLK-KLWMIJ hamiltonian is not self dual, the remnant of this symmetry survives on the zero eigenvalue subspace. The full selfdual RFT Hamitonian (which we will refer to as H RF T ) is not known yet, although a number of selfdual extensions of the JIMWLK evolution has been proposed and derived in some approximations [26,27,28] (see however [29]). …”