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We propose a stochastic particle model in (1+1)-dimensions, with one dimension corresponding to rapidity and the other one to the transverse size of a dipole in QCD, which mimics high-energy evolution and scattering in QCD in the presence of both saturation and particle-number fluctuations, and hence of Pomeron loops. The model evolves via non-linear particle splitting, with a non-local splitting rate which is constrained by boost-invariance and multiple scattering. The splitting rate saturates at high density, so like the gluon emission rate in the JIMWLK evolution. In the mean field approximation obtained by ignoring fluctuations, the model exhibits the hallmarks of the BK equation, namely a BFKL-like evolution at low density, the formation of a traveling wave, and geometric scaling. In the full evolution including fluctuations, the geometric scaling is washed out at high energy and replaced by diffusive scaling. It is likely that the model belongs to the universality class of the reaction-diffusion process. The analysis of the model sheds new light on the Pomeron loops equations in QCD and their possible improvements.
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We construct the effective Hamiltonian which governs the renormalization group flow of the gluon distribution with increasing energy and in the leading logarithmic approximation. This Hamiltonian defines a two-dimensional field theory which involves two types of Wilson lines: longitudinal Wilson lines which describe gluon recombination (or merging) and temporal Wilson lines which account for gluon bremsstrahlung (or splitting). The Hamiltonian is self-dual, i.e., it is invariant under the exchange of the two types of Wilson lines. In the high density regime where one can neglect gluon number fluctuations, the general Hamiltonian reduces to that for the JIMWLK evolution. In the dilute regime where gluon recombination becomes unimportant, it reduces to the dual partner of the JIMWLK Hamiltonian, which describes bremsstrahlung.
We calculate the partition functions of QCD in two dimensions on a cylinder and on a torus in the gauge $partial_{0} A_{0} = 0$ by integrating explicitly over the non zero modes of the Fourier expansion in the periodic time variable. The result is a one dimensional Kazakov-Migdal matrix model with eigenvalues on a circle rather than on a line. We prove that our result coincides with the standard expansion in representations of the gauge group. This involves a non trivial modular transformation from an expansion in exponentials of $g^2$ to one in exponentials of $1/g^2$. Finally we argue that the states of the $U(N)$ or $SU(N)$ partition function can be interpreted as a gas of N free fermions, and the grand canonical partition function of such ensemble is given explicitly as an infinite product.
We show that the recently developed Hamiltonian theory for high energy evolution in QCD in the dilute regime and in the presence of Bremsstrahlung is consistent with the color dipole picture in the limit where the number of colors N_c is large. The color dipoles are quark-antiquark pairs which can radiate arbitrarily many soft gluons, and the evolution consists in the splitting of any such a dipole into two. We construct the color glass weight function of an onium as a superposition of color dipoles, each represented by a pair of Wilson lines. We show that the action of the Bremsstrahlung Hamiltonian on this weight function and in the large-N_c limit generates the evolution expected from the dipole picture. We construct the dipole number operator in the Hamiltonian theory and deduce the evolution equations for the dipole densities, which are again consistent with the dipole picture. We argue that the Bremsstrahlung effects beyond two gluon emission per dipole are irrelevant for the calculation of scattering amplitudes at high energy.
We use a renormalization group method to treat QCD-vacuum behavior specially closer to the regime of asymptotic freedom. QCD-vacuum behaves effectively like a paramagnetic system of a classical theory in the sense that virtual color charges (gluons) emerges in it as a spin effect of a paramagnetic material when a magnetic field aligns their microscopic magnetic dipoles. Due to that strong classical analogy with the paramagnetism of Landaus theory,we will be able to use a certain Landau effective action without temperature and phase transition for just representing QCD-vacuum behavior at higher energies as being magnetization of a paramagnetic material in the presence of a magnetic field $H$. This reasoning will allow us to apply Thompsons approach to such an action in order to extract an effective susceptibility ($chi>0$) of QCD-vacuum. It depends on logarithmic of energy scale $u$ to investigate hadronic matter. Consequently we are able to get an ``effective magnetic permeability ($mu>1$) of such a paramagnetic vacuum. Actually,as QCD-vacuum must obey Lorentz invariance,the attainment of $mu>1$ must simply require that the effective electrical permissivity is $epsilon<1$ in such a way that $muepsilon=1$ ($c^2=1$). This leads to the anti-screening effect where the asymptotic freedom takes place. We will also be able to extend our investigation to include both the diamagnetic fermionic properties of QED-vacuum (screening) and the paramagnetic bosonic properties of QCD-vacuum (anti-screening) into the same formalism by obtaining a $beta$-function at 1 loop,where both the bosonic and fermionic contributions are considered.
We study the high density region of QCD within an effective model obtained in the frame of the hopping parameter expansion and choosing Polyakov type of loops as the main dynamical variables representing the fermionic matter. To get a first idea of the phase structure, the model is analyzed in strong coupling expansion and using a mean field approximation. In numerical simulations, the model still shows the so-called sign problem, a difficulty peculiar to non-zero chemical potential, but it permits the development of algorithms which ensure a good overlap of the Monte Carlo ensemble with the true one. We review the main features of the model and present calculations concerning the dependence of various observables on the chemical potential and on the temperature, in particular of the charge density and the diquark susceptibility, which may be used to characterize the various phases expected at high baryonic density. We obtain in this way information about the phase structure of the model and the corresponding phase transitions and cross over regions, which can be considered as hints for the behaviour of non-zero density QCD.
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