Conformal symmetry of JIMWLK evolution at NLO
Alex Kovner
1
Michael Lublinsky
0
Yair Mulian
0
0
Physics Department, Ben-Gurion University of the Negev
, Beer Sheva 84105,
Israel
1
Physics Department, University of Connecticut
, 2152 Hillside road, Storrs,
CT 06269, U.S.A
We construct the Next to Leading Order JIMWLK Hamiltonian for high energy evolution in N = 4 SUSY theory, and show that it possesses conformal invariance, even though it is derived using sharp cutoff on rapidity variable. The conformal transformation properties of Wilson lines are not quite the naive ones, but at NLO acquire an additional anomalous piece. We construct explicitly the inversion symmetry generator. We also show how to construct for every operator O, including the Hamiltonian itself, its conformal extension O, such that it transforms under the inversion in the naive way.
Contents
1 Introduction
2 The NLO JIMWLK Hamiltonian.
3 Naive conformal transformations.
4 The Conformal Symmetry of the Hamiltonian.
5 Constructing conformal operators.
6 Conclusions
QCD NLO JIMWLK Hamiltonian
B Technical details of derivations
B.1 Action of the NLO JIMWLK Hamiltonian on the dipole
B.2 Algebra with kernels.
1 Introduction
In recent years a lot of attention has been devoted to development and phenomenological
applications of the theory of perturbative saturation [1]. The main physical idea of this
approach is that at high enough energy hadronic wave function resembles a dense gluon
cloud, sometimes referred to as the Color Glass Condensate (CGC) [24]. When the
energy is high enough the density scale becomes large and the physics becomes essentially
perturbative and tractable.
The theoretical description of the energy evolution of the wave function towards such
a dense state at leading order in s has been long known. It is given by the so called
JIMWLK equation [210], or equivalently Balitsky hierarchy [1113]. It generalizes the
well known BFKL equation [1418] by including finite density effects in the hadronic wave
function.
The mean field approximation to the JIMWLK equation, the so called
BalitskyKovchegov (BK) equation [1113, 19] has been used extensively in the last several years
in phenomenological applications, that include fits to DIS low x data [20, 21] as well as
various aspects of p-p and p-A data [22]. For phenomenological applications it is crucial
to include perturbative corrections beyond leading order, since they are known to lead to
large effects already in the linear BFKL framework [23, 24]. Currently only the corrections
due to running coupling constant ([2528]) are included in the numerical work, although
there has been recent progress in understanding of the more problematic gluon emission
contributions [29].
Significant progress in including the full set of next to leading order corrections in
the high energy evolution was achieved by Balitsky and Chirilli [30]. This work presented
the complete set of NLO corrections to the evolution of the scattering amplitude of the
fundamental dipole in QCD. Subsequently analogous calculation was performed in the
N = 4 super Yang Mills theory [31]. Recently Grabovsky [32] computed certain parts of
the NLO evolution equation for three-quark singlet amplitude in the SU(3) theory.
Using the results of [30] and [32] in a recent paper [33] we have derived the complete
operator form of the JIMWLK Hamiltonian at next to leading order. This paper appeared
simultaneously with [34], which directly calculated most elements of the general Balitsky
hierarchy at NLO. We also note that similar results have been independently obtained
in [35]. For the sake of self-completeness of the present paper, in appendix A we quote
the main result of ref. [33]. In appendix B we provide some insight on how the result is
obtained, while a more detailed report of our derivation and a comparison with [34] will
appear in a forthcoming separate publication [36].
Even though the JIMWLK NLO Hamiltonian is now available, there are theoretical
questions about it that still have to be addressed. In this paper we address one such
question, namely that of conformal invariance. The leading order JIMWLK equation is
conformally invariant when applied on gauge invariant states. This holds even in QCD,
although conformal invariance of the classical Yang-Mills action is violated by the quantum
anomaly, since the coupling constant renormalization is necessary only beyond leading
order.
The NLO evolution of the color dipole derived in [30] as well as the explicit form of
the NLO JIMWLK Hamiltonian given in [33] is not conformally invariant. There are two
sources for the violation of conformal symmetry at this level. One is due to the genuine
quantum anomaly associated with the introduction of renormalization scale. The other one
is due to the fact that the calculations involve hard cutoff in rapidity space, which itself
is not conformally invariant [37]. In principle it should be possible to eliminate this latter
source of noninvariance by employing an explici (...truncated)