The Jalilian-Marian,Iancu, McLerran, Weigert, Leonidov, Kovner (JIMWLK) Hamiltonian for high energy evolution of QCD amplitudes is presented at the next-to-leading order accuracy in αs. The form of the Hamiltonian is deduced from the symmetries and the structure of the hadronic light cone wavefunction and by comparing the rapidity evolution of the quark dipole and the three-quark singlet states with results available in the literature. The next-to-leading corrections should allow for more robust phenomenological applications of perturbative saturation approach.It is believed that at high energy gluons saturate in perturbative regime. The idea of perturbative gluon saturation was first suggested and discussed in detail in [1]. To date there exist numerous phenomenological applications of this idea to DIS, heavy ion collisions and proton-proton collisions at the LHC [2]. These applications are based on the Balitsky-Kovchegov (BK) non-linear evolution equation [3,4], which at large N c describes the growth of the gluon density with energy and the gluon saturation. The more general approach to the calculation of high energy hadronic amplitudes is known as the Jalilian-Marian,Iancu, McLerran, Weigert, Leonidov, Kovner (JIMWLK) evolution. The JIMWLK Hamiltonian [5] is the limit of the QCD Reggeon Field Theory (RFT), applicable for computations of high energy scattering amplitudes of dilute (small parton number) projectiles on dense (nuclei) targets. In general it predicts the rapidity evolution of any hadronic observable O via the functional equation of the formIn ref.[5], the JIMWLK Hamiltonian was derived in the leading order in α s in pQCD. It contains a wealth of information about high energy evolution equations. In the dilute-dilute limit it generates the linear BFKL equation [6] and its BKP extension [7]. Beyond the dilute limit, the Hamiltonian incorporates non-linear effects responsible for unitarization of scattering amplitudes. The BK equation arises as the mean field approximation to JIMWLK evolution at large N c . Successful BK phenomenology mandates inclusion of next to leading order corrections, since at leading order the evolution predicted by the BK equation is too rapid to describe experimental data. Currently only the running coupling corrections are included in applications, although it is clearly desirable to include all next to leading corrections. The complete set of such corrections to the evolution of a fundamental dipole was calculated in a remarkable paper by Balitsky and Chirilli [8]. The complete functional JIMWLK equation however, at the moment is only known at leading order. The next to leading order extension of the JIMWLK framework is imperative for calculation of more general amplitudes, beyond the dipole, which determine interesting experimental observables like single-and double inclusive particle production [9].Beyond phenomenological interest, deriving and exploring RFT at NLO is a fundamental theoretical question and it is the focus of this paper.The NLO BFKL equation was derived ...

We derive corrections to the JIMWLK equation in the regime where the charge density in the hadronic wave function is small. We show that the framework of the JIMWLK equation has to be significantly modified at small densities in order to properly account for the noncommutativity of the charge density operators. In particular the weight function for the calculation of averages can not be real, but is shown to contain the Wess-Zumino term. The corrections to the kernel of the JIMWLK evolution which are leading at small density are resummed into a path ordered exponential of the functional derivative with respect to the charge density operator, thus hinting at intriguing duality between the high and the low density regimes.

We prove that the complete kernel for the high-energy evolution in QCD must be self-dual. The relevant duality transformation is formulated in precise mathematical terms and is shown to transform the charge density into the functional derivative with respect to the single-gluon scattering matrix. This transformation interchanges the high and the low density regimes. We demonstrate that the original Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner kernel, valid at large density, is indeed dual to the low density limit of the complete kernel derived recently in hep-ph/0501198.

We discuss several issues related to recent work on high energy evolution. In particular we show that the Hilbert space for action of the operator of the evolution can be conveniently defined by the space of weight functional introduced recently in [28]. These weight functionals automatically implement the correct properties of the charge density correlators, thus eliminating the need in explicit introduction of the Wess Zumino term. We also discuss various aspects of Dense Dilute Duality in the toy dipole model.

We present a general, model independent argument demonstrating that gluons produced in high energy hadronic collision are necessarily correlated in rapidity and also in the emission angle. The strength of the correlation depends on the process and on the structure/model of the colliding particles. In particular we argue that it is strongly affected (and underestimated) by factorized approximations frequently used to quantify the effect.