Nonlinear evolution equations in QCD and effective Hamiltonian at high energy

In this talk I briefly present recent developments in the theory of the Color Glass Condensate. The duality between the dense and dilute regimes of the gluon field is discussed as well as the effective selfdual Hamiltonian which includes both Pomeron merging and Pomeron splitting. Introduction: high energy limit and the Pomeron One of the central problems in the theory of strong interactions is the understanding of the behaviour of the hadronic cross sections in the limit of high energies. The experimental data on the total cross section show slow but distinct increase with energy. It has been suggested that this rise, which can be parametrized by a power-like form, σ(s) ∼ s0.08 , (1) is mediated by the ’soft Pomeron’ [1]. Perturbative calculation summing the leading logarithmic contributions of large logs lns/t results in the BFKL Pomeron, which however leads to a stronger rise of the cross section with energy [2] σ(s) ∼ s4ln2αsNc/π , (2) with the leading exponent being much larger than in (1) when evaluated at physically interesting values of the strong coupling constant (αs ≃ 0.2). In any of these cases, the power-like increase of the cross section is in contradiction with the Froissart bound [3] which allows at most logarithmic increase with energy σ(s) ∼ σ0 ln2 s , (3) with a normalization coefficient σ0 related to the inverse of the pion mass squared. Froissart bound stems from the very general principles: unitarity and the finite range of the strong interactions. Thus, the important question arises whether it is possible to identify and calculate other type of Feynman diagrams which will lead to the restoration of the unitarity. Color Glass Condensate and the JIMWLK equation The Color Glass Condensate is an effective theory of the strong interactions at very high energies. Apart from the standard BFKL evolution of the gluon density it also contains the recombination diagrams which are important when the gluon density becomes very high. It therefore describes the phenomenon known as perturbative parton saturation. When the density of gluons becomes very high due to their enhanced splitting described by the BFKL Pomeron, the recombination effects start to become important and reduce the growth. It is believed that this phenomenon is important for the restoration of the unitarity. The Color Glass Condensate theory describes the scattering of the projectile off a target which constitutes a dense system of soft gluons. The S-matrix for the scattering of the quark-antiquark dipole in the target field α is described by S(x,y) = 1 Nc Tr(VxV † y ) , (4) where Wilson line Vx = Pexp ( ig ∫ dxαa(x,x) ta ) , (5) is the path ordered exponential along the trajectory of the projectile. The physical scattering matrix between the qq̄ dipole and the hadron target is obtained by taking the average of (4) over the color field of the target 〈S(x,y)〉τ = ∫ D[ρ]Z [ρ ]τ S(x,y) , (6) where ρ is a color charge which generates field α . One cannot compute the weight function Z [ρ] since it contains nonperturbative information about the hadron but one can compute the evolution of this weight function with increasing rapidity τ ∼ lns. The basic equation of the Color Glass Condensate theory is the JIMWLK equation [4] which governs the evolution of the weight function Z [ρ]τ with rapidity ∂Z [ρ]τ ∂τ = HJIMWLK Z [ρ]τ , (7) where H is the JIMWLK Hamiltonian. The JIWMLK equation is a very complicated functional evolution equation, which however reduces to one closed and relatively simple equation, the Balitsky-Kovchegov equation, in the large Nc limit and in the dipole picture. Diagramatically JIMWLK equation contains BFKL Pomeron ladder diagrams as well as the triple Pomeron interaction diagrams which reduce the growth of the gluon density. More precisely it contains Pomeron merging diagrams, however it misses the Pomeron splittings. One can understand it by looking at the structure of the operator HJIMWLK HJIMWLK = 1 2π ∫ x,y,z Kxyz δ δαa(x) [ 1+Ṽ † x Ṽy −Ṽ † z Ṽy −Ṽ † x Ṽz ]ab δ δαb(y) , (8) which is second order in derivatives δ δα and all orders (through Ṽ ’s) 1 in the field α . That means, that the evolution of the correlation functions in the field α can be (by using (6,7,8)) schematically represented as ∂ n { }} { 〈α . . .α〉 ∂τ = ∑ m≥n Km⊗ m { }} { 〈α . . .α〉 . (9) The evolution of n point functions is coupled to the higher order correlation functions but not to the lower correlation functions. Dual version of JIMWLK and effective Hamiltonian In order to include the additional diagrams corresponding to the Pomeron splittings in the Color Glass Condensate formalism, terms which are higher order in derivatives have to be incorporated [5]. By including these terms in the evolution, the Pomeron loops are generated, which are known to be important contributions at very high energy. It has been suggested [6], that in general there is a duality relation between the evolution of the dense system, governed by the JIWMLK equation, and the dilute regime. In particular, the Pomeron mergings and splittings are believed to be the dominant diagrams in the dense and the dilute regimes, respectively, of the gluon field. In particular the evolution in the dilute regime is governed by the dual equation to the JIMWLK evolution. Formally, the duality can be expressed as the transformation [6, 7] x ↔ x, δ δαa(x−,x) ↔ iρ(x,x), α(x,x) ↔−i δ δρa(x+,x) , (10) where ρ is the charge of the target and the α is the field generated by this charge measured by the projectile. The dual version of the JIMWLK equation is then ∂Z [ρ]τ ∂τ = H̃ Z [ρ]τ , (11) where the dual Hamiltonian