abstract

At high energies, the properties of hadronic and nuclear interactions are determined by the dynamics of strong color fields. The physics of strong color fields is responsible for many of the phenomena observed at RHIC and is expected to describe both \(pp\) and \(AA\) collisions at the LHC. These lectures are intended as an elementary introduction into QCD and its behavior in the strong field limit.

abstract

I argue that the physics of the scattering of very high energy strongly interacting particles is controlled by a new, universal form of matter, the Color Glass Condensate. This matter is the dominant contribution to the low \(x\) part of a hadron wavefunction. In collisions, this mater almost instantaneously turns into a Glasma. The Glasma initially has strong longitudinal color electric and magnetic fields, with topological charge. These fields melt into gluons. Due to instabilities, quantum noise is converted into classical turbulence, which may be responsible for the early thermalization seen in heavy ion collisions at RHIC.

abstract

In the Color Glass Condensate (CGC) effective field theory, when two large sheets of Colored Glass collide, as in a central nucleus–nucleus collision, they form a strongly interacting, non-equilibrium state of matter called the Glasma. How Colored Glass shatters to form the Glasma, the properties of the Glasma, and the complex dynamics transforming the Glasma to a thermalized Quark Gluon Plasma (QGP) are questions of central interest in understanding the properties of the strongly interacting matter produced in heavy ion collisions. In the first of these lectures, we shall discuss how these questions may be addressed in the framework of particle production in a field theory with strong time dependent external sources. Albeit such field theories are non-perturbative even for arbitrarily weak coupling, moments of the multiplicity distribution can in principle be computed systematically in powers of the coupling constant. We will demonstrate that the average multiplicity can be (straightforwardly) computed to leading order in the coupling and (remarkably) to next-to-leading order as well. The latter are obtained from solutions of small fluctuation equations of motion with retarded boundary conditions. In the second lecture, we relate our formalism to results from previous 2+1 and 3+1 dimensional numerical simulations of the Glasma fields. The latter show clearly that the expanding Glasma is unstable; small fluctuations in the initial conditions grow exponentially with the square root of the proper time. Whether this explosive growth of small fluctuations leads to early thermalization in heavy ion collisions requires at present a better understanding of these fluctuations on the light cone. In the third and final lecture, motivated by recent work A. Bialas, M. Jeżabek, Phys. Lett. B590, 233 (2004), we will discuss how the widely observed phenomenon of limiting fragmentation is realized in the CGC framework.

abstract

Is the new state of matter formed in relativistic heavy ion collisions the deconfined quark–gluon plasma? We survey the status of several strange hadron observables and discuss how these measurements help understand the dense hadronic matter.

abstract

The principal problems of the matter evolution in hydrodynamic picture of \(A+A\) collisions are considered. They concern the process of freeze-out, possible duality in the hydrodynamic and kinetic descriptions, and formation of the initial conditions at pre-thermal partonic stage, in particular, developing of initial transverse velocities and angular momentum. Also time evolution of the momentum spectra, Bose–Einstein correlation functions and averaged phase-space densities (APSD) are analyzed within hydrodynamic and kinetic models. The results shed light on the behavior of the observables, in particular, the interferometry volumes and APSD in \(A+A\) collisions at different energies and for different nuclei. The new class of freeze-out parameterizations, accounting for continuous in time particle emission, is considered and their hydrodynamical realization is discussed.

abstract

After a brief overview of the high energy heavy ion collisions and lattice QCD simulations at finite temperature and density, we report our lattice study of transport coefficients of quark–gluon plasma (QGP) in the lattice quench approximation. We discuss why the transport coefficients are important and interesting in QGP physics.

abstract

The effects of shape of the freeze-out hypersurface and resonance decays on the pion correlation functions in relativistic heavy-ion collisions are studied with help of the hydro-inspired models with single freeze-out. The heavy-ion Monte Carlo generator THERMINATOR is used to generate hadronic events describing production of particles from a thermalized and expanding source. We find that the short-lived resonances increase the pionic HBT radii by about 1 fm. We also find that the pion HBT data from RHIC are fully compatible with the single freeze-out scenario provided a special choice of the freeze-out hypersurface is made.

abstract

We show that the boost-invariant and cylindrically asymmetric hydrodynamic equations for baryon-free matter may be rewritten as only two coupled partial differential equations. In the case where the system exhibits the cross-over phase transition, the standard numerical methods may be applied to solve these equations. An example of our results describing non-central gold on gold collisions at RHIC energies is presented.

abstract

Thermodynamic quantities in symmetric nuclear matter are calculated in a self-consistent \(T\)-matrix approximation at zero and finite temperatures. The internal energy is calculated from the Galitskii–Koltun’s sum rule and from the summation of the diagrams for the interaction energy. The pressure and the entropy at finite temperature are obtained from the generating functional form of the thermodynamic potential.

abstract

The final \(p_{\rm T}\)-spectra measured at RHIC at \(\sqrt {s_{NN}}=130\) and 200 GeV are fitted within the Cracow single-freeze-out model. Then the global variables like the transverse energy at midrapidity, the charged particle multiplicity at midrapidity and the total multiplicity of charged particles are evaluated. The predictions agree fairly well with the experimental data. The centrality independence of the total number of charged particles per participant pair has been also reproduced.

abstract

A concept of wounded nucleons and/or wounded quarks plays an important role in parametrizing and to some extent explaining many a feature of the relativistic ion collisions. This will be illustrated in a historical perspective, up to and including the latest developments.

abstract

The NA49 fixed-target experiment studied high energy-density matter produced in nucleus–nucleus reactions at the CERN SPS. In central Pb+Pb collisions at 158\(A\) GeV the energy density at the early stage substantially exceeds the threshold for quark deconfinement predicted by lattice QCD. The produced matter shows strong transverse and longitudinal flow. Ratios of yields of produced particles are approximately consistent with statistical equilibration. An energy scan through the SPS range revealed structure in the energy dependence of \(\pi \) and \(K\) yields as well as of the inverse slopes of transverse mass distributions. These features suggest that a deconfined phase starts to be produced at around 30\(A\) GeV in central Pb+Pb collisions. The analysis of fluctuations and correlations has not yet provided evidence for the predicted critical point of QCD.

abstract

High energy scattering was recently shown to be similar to a reaction-diffusion process. The latter defines a wide universality class that also contains e.g. some specific population evolution models. The common point of all these models is that their respective dynamics are described by noisy traveling wave equations. This observation has led to a new understanding of QCD in the regime of high energies, and known universal results on reaction-diffusion models could be transposed to obtain quantitative properties of QCD amplitudes. Conversely, new general results for that kind of statistical models have also been derived. Furthermore, an intriguing relationship between noisy traveling wave equations and the theory of spin glasses was found.

abstract

In this artcle, I shall review the basic concepts of perturbative QCD in its high-energy limit. I shall concentrate on the approach to the unitarity limit, usually referred to as saturation, as well as on the gluon-number fluctuations the importance of which has recently been discovered. I shall explain the basic framework showing the need for those phenomena, first, from a simple picture of the high-energy behaviour, then, giving a short derivation of the equation driving this evolution. In the second part, I shall exhibit an analogy with statistical physics and show how this allows to derive geometric scaling in QCD with saturation. I shall finally consider the effects of gluon-number fluctuations on this picture and draw the physical consequences, i.e. a new scaling law, arising from those results.

abstract

When quarks and gluons tend to form a dense medium, like in high energy or/and heavy-ion collisions, it is interesting to ask the question which are the relevant degrees of freedom that Quantum Chromodynamics predicts. The present notes correspond to two lectures given at Zakopane in the (rainy) summer of 2006, where this question is adressed concretely in two cases, one in the QCD regime of weak coupling, the other one at strong coupling. (The second Lecture is published on page 3525 of this volume.) Each case corresponds to the study of an elusive but dynamically important transient phase of quarks and gluons expected to appear from Quantum Chromodynamics during high energy collisions. Below, we examine the dynamical phase space of gluon transverse momenta near the so-called “saturation” phase including its fluctuation pattern. “Saturation” is expected to appear when the density of gluons emitted during the collision reaches the limit when recombination effects cannot be neglected, even in the perturbative QCD regime. We demonstrate that the gluon momenta exhibit a nontrivial clustering structure, analoguous to “hot spots”, whose distributions are derived using an interesting matching with the thermodynamics of directed polymers on a tree with disorder and its “spin-glass” phase.

abstract

We discuss the non-perturbative regime of QCD, which is supposed to be relevant for the description of the transient phase as quark–gluon plasma formed during heavy-ion collisions at very high energies. Since there is not yet an available field-theoretical scheme for non perturbative QCD in those conditions, we study the dynamics of strongly interacting gauge-theory matter (modelling quark–gluon plasma) using the AdS/CFT duality between gauge field theory at strong coupling and a gravitational background in Anti-de Sitter space. The relevant gauge theory is a-priori equipped with \({\cal N}=4\) supersymmetries, but qualitative results may give lessons on this issue. As an explicit example, we show that perfect fluid hydrodynamics emerges at large times as the unique nonsingular asymptotic solution of the nonlinear Einstein equations in the bulk. The gravity dual can be interpreted as a black hole moving off in the fifth dimension.

abstract

An approach to the gluon saturation is discussed within a framework of interacting QCD pomeron field theory. The formulation is consistent with Lorentz invariance which guarantees that the symmetry between the target and the projectile of the scattering matrix is preserved. The dynamics of interacting pomeron system is studied in the semi-classical approximation. Solutions to the emerging classical equations of motion (Braun equations) are presented. Two unexpected features of these solutions are found: a break-down of the symmetry between the target and the projectile and their similarity to solutions of the Balitsky–Kovchegov equation. Interpretation of the results is given and possible consequences are shortly discussed.

abstract

In the context of both inclusive and diffractive deep inelastic scattering, we derive the first phenomenological consequences of the inclusion of Pomeron loops in the QCD evolution equations towards high-energy. We discuss the transition between the well-known geometric scaling regime and the new diffusive scaling, that emerges for sufficiently high energies and up to very large values of \(Q^2,\) well above the proton saturation momentum.

abstract

We present a formalism which modifies the Mueller Dipole Model such that it incorporates energy-momentum conservation and also important colour suppressed effects. We implement our formalism in a Monte Carlo simulation and compare the results to inclusive data from HERA and the Tevatron, where we see that there is a good agreement between the data and our model.

abstract

In these lectures, we present and discus the most recent results on inclusive diffraction from the HERA and Tevatron colliders and give the prospects for the future at the LHC. Of special interest is the exclusive production of Higgs boson and heavy objects (\(W\), top, stop pairs) which will require a better understanding of diffractive events and the link between \(ep\) and hadronic colliders, as well as precise measurements and analyses of inclusive diffraction at the LHC in particular to constrain further the gluon density in the pomeron.

abstract

We illustrate the general scheme of the Sum Rule (SR) method using 2D Quantum Harmonic Oscillator (2DQHO) as a toy model. We introduce correlator, related to Green function of 2DQHO, and describe the property of Asymptotic Freedom for 2DQHO. We explain how the duality conception allows one to describe excited states. Finally we present numerical results and extract some lessons to learn from our exposition. Then we switch to the QCD and show that QCD SRs supply us the method to study hadrons in non-perturbative QCD.

abstract

We show that the minimal Gaussian model of nonlocal vacuum quark and quark–gluon condensates in QCD generates the non-transversity of vector current correlators. We suggest the improved Gaussian model of the nonperturbative QCD vacuum, which respects QCD equation of motion and minimizes the revealed gauge-invariance breakdown. We obtain the refined values of pion distribution amplitude (DA) conformal moments \(\langle \xi ^{2N}\rangle _{\pi } (N=1,\) … , 5) using the improved QCD vacuum model, including the inverse moment \(\langle x^{-1}\rangle _{\pi }\), being inaccessible if one uses the standard QCD sum rules. We construct the allowed region for Gegenbauer coefficients \(a_2\) and \(a_4\) of the pion DA for two values of the QCD vacuum nonlocality parameter, \(\lambda _q^2=0.4\) and 0.5 GeV\(^2\).

abstract

After summarizing briefly some numerical results for four-dimensional supersymmetric SU(2) Yang–Mills quantum mechanics, we review a recent study of systems with an infinite number of colours. We study in detail a particular supersymmetric matrix model which exhibits a phase transition, strong-weak duality, and a rich structure of supersymmetric vacua. In the planar and strong coupling limits, this field theoretical system is equivalent to a one-dimensional XXZ Heisenberg chain and, at the same time, to a gas of \(q\)-bosons. This not only reveals a hidden supersymmetry in these well-studied models; it also maps the intricate pattern of our supersymmetic vacua into that of the now-popular ground states of the XXZ chain.

abstract

Supersymmetric Yang–Mills quantum mechanics (SYMQM) in four dimensions for SU(2) gauge group is considered. In this work a two-fermionic sector with the angular momentum \(j=0\) in discussed. Energy levels from discrete and continuous spectra are calculated. To distinguish localized states from non-localized ones the virial theorem is applied.

abstract

We briefly present a relationship between General Relativity coupled to certain spin-0 and spin-2 field theories and higher derivatives metric theories of gravity. In a special case, described by the Einstein–Bach equations, the spin-0 field drops out from the theory and we obtain a consistent spin-two field theory interacting gravitationally, which overcomes a well known inconsistency of the theory for a linear spin-two field coupled to the Einstein’s gravity. Then we discuss basic properties of static spherically symmetric solutions of the Einstein–Bach equations.