We show that for a finite number of emitting sources, the Color Glass Condensate produces substantial elliptic azimuthal anisotropy, characterized by $v_2$, for two and four particle correlations for momentum greater than or of the order the saturation momentum. The flow produced has the correct semi-quantitative features to describe flow seen in LHC experiments with p-Pb and pp collisions. This flow is induced by quantum mechanical interference between the waves of produced particles, and the flow itself is coupled to fluctuations in the positions of emitting sources. We shortly discuss generalizing these results to odd $v_n$, to correlations involving larger number of particles, and to transverse momentum scales $Λ_{\rm QCD} \ll p_T \ll Q_{\rm sat}$.
I introduce the concept of the Color Glass Condensate. I review data from HERA and RHIC which suggest that such a universal form of matter has been found.
I briefly review the physical picture of the saturated gluons at small-x as a Colour Glass Condensate, and the effective theory which forms the basis of this picture.
The Color Glass Condensate provides a systematic power counting of initial state effects in high energy QCD. We briefly discuss in this framework quark and gluon production in high energy collisions.
We explain the main ideas of the color glass condensate in high energy collisions. Different approaches to the problem are outlined with emphasis on the resummation approach. We present evidence that the color glass condensate can be described by an effective conformal field theory or even by a string theory.
We systematically compute the Gaussian average of Wilson lines inherent in the Color Glass Condensate, which provides useful formulae for evaluation of the scattering amplitude in the collision of a light projectile and a heavy target.
These two lectures concern the Color Glass Condensate and the Glasma. These are forms of matter which might be studied in high energy hadronic collisions. The Color Glass Condensate is high energy density gluonic matter. It constitutes the part of a hadron wavefunction important for high energy processes. The Glasma is matter produced from the collision of two high energy hadrons. Both types of matter are associated with coherent fields. The Color Glass Condensate is static and related to a hadron wavefunction where the glasma is transient and evolves quickly after a collision. I present the p…
We make quantitative predictions for the rapidity and centrality dependencies of hadron multiplicities in $AA$, $pA$ and $pp$ collisions at the LHC energies basing on the ideas of parton saturation in the Color Glass Condensate.
A model for the Color Glass Condensate as opposed to jet quenching is proposed for the explanation of the presently available RHIC data. Good fits to these data are presented. A clear way to distinguish between the two possible explanations is also given.
The Color Glass Condensate is a theory of the dynamical properties of partons in the Regge limit of QCD: x_{\rm Bj}\to 0$, $Q^2 >> Λ_{\rm QCD}^2={\rm fixed} and the center of mass energy squared s\to \infty. We provide a brief introduction to the theoretical ideas underlying the Color Glass Condensate and discuss the application of these ideas to high energy scattering in QCD.
We give an elementary discussion of parton saturation and its description by the effective theory of the Color Glass Condensate. We report on progress in calculating multi-gluon correlators. The latter are necessary for many phenomenological applications, upon some of which we briefly touch.
The Color Glass Condensate is a state of high density gluonic matter which controls the high energy limit of hadronic matter. These lectures begin with a discussion of general problems of high energy strong interactions. The infinite momentum frame description of a single hadron at very small x is developed, and this picture is applied to the description of ultrarelativistic nuclear collisions. Recent developments in the renormalization group description of the Color Glass Condensate are reviewed.
This contribution discusses the phenomenon of parton saturation, the color glass picture of hadronic wavefuntions, and their relevance in the early stages of nucleus-nucleus collisions. Evidence for the color glass condensate in the presently available RHIC data is critically reviewed.
I review the problem of parton saturation and its implications through three in principal different approaches, but somewhat related: saturation in a geometrical approach, QCD saturation through the Color Glass Condensate and shadowing in a Pomeron approach.
We discuss general properties of the Color Glass Condensate. We show that predictions for particle production in p(d)A and AA collisions derived from these properties are in agreement with data collected at RHIC.
We confront the colour glass condensate motivated dipole model parameterization of Iancu, Itakura and Munier with the available HERA data on the diffractive structure function F2D3 and with existing dipole model parameterizations. Reasonably good agreement is found with only two adjustable parameters. We caution against interpreting the success of the model as compelling evidence for low-x perturbative saturation dynamics.
We discuss forward-backward correlations in the mutliplicity of produced particles in heavy ion collisions. We find the Color Glass Condensate generates distinctive predictions for the long range component of this correlation. In particular, we predict the growth of the long range correlation with the centrality of the collision. We argue that the correlation for mesons is stronger than that for baryons.
The high energy limit of QCD is controlled by the small-$x$ part of a hadron wavefunction. I argue that this part is universal to all hadrons and is composed of a new form of matter: a Colored Glass Condensate. This matter is weakly interacting at very small $x$, but is non-perturbative because of the highly occupied boson states which compose the condensate. Such a matter might be studied in high energy lepton-hadron or hadron-hadron interactions.
The proton structure function F_2(x,Q^2) for x < 0.01 and 0.045< Q^2 < 45 GeV^2, measured in the deep inelastic scattering at HERA, can be well described within the framework of the Color Glass Condensate.
The purpose of these lectures is to provide an introduction to the physics issues which are being studied in the RHIC heavy ion program. These center around the production of new states of matter. The Quark Gluon Plasma is thermal matter which once existed in the big bang which may be made at RHIC. The Color Glass Condensate is a universal form of matter which controls the high energy limit of strong interactions. Both such forms of matter might be produced and probed at RHIC.
In these lectures, we develop the theory of the Colour Glass Condensate. This is the matter made of gluons in the high density environment characteristic of deep inelastic scattering or hadron-hadron collisions at very high energy. The lectures are self contained and comprehensive. They start with a phenomenological introduction, develop the theory of classical gluon fields appropriate for the Colour Glass, and end with a derivation and discussion of the renormalization group equations which determine this effective theory.
Multi-particle production in QCD is dominated by higher twist contributions. The operator product expansion is not very effective here because the number of relevant operators grow rapidly with increasing twist. The Color Glass Condensate (CGC) provides a framework in QCD to systematically discuss "classical" (multiple scattering) and "quantum" evolution (shadowing) effects in multi-particle production. The apparently insuperable problem of nucleus-nucleus scattering in QCD simplifies greatly in the CGC. A few examples are discussed with emphasis on open problems.
We discuss proton-nucleus collisions in the framework of the color glass condensate. By assuming that the proton can be described as a low density color source, we solve exactly the Yang-Mills equations corresponding to this type of collision, and then use this solution in order to calculate inclusive gluon production or quark-antiquark production. Our result shows that k_T-factorization, while valid for gluon production, is violated for quark pair production in proton-nucleus collisions.
Following the Good-and-Walker picture, hard diffraction in the high-energy/small-x limit of QCD can be described in terms of eigenstates of the scattering matrix off a Color Glass Condensate. From the CGC non-linear evolution equations, it is then possible to derive the behavior of diffractive cross-sections at small $x.$ I discuss recent results, in particular the consequences of the inclusion of Pomeron loops in the evolution.
The Color Glass Condensate is a universal state of matter which can manifest itself in hadronic processes involving small-x partons, like DIS and pp, pA and AA collisions at high energy. Observables are given in terms of multi-gluon correlators, whose ensemble evolves according to a RG equation, the JIMWLK equation. We focus on recent progress towards its solution which lead to quasi-exact, analytic expressions for the multi-gluon correlators at high energy.
We discuss heavy quark production in High Parton Density QCD in quasi-classical approximation and including low-$x$ quantum evolution. We also consider an alternative approach based on the effect of pair production in external fields.
We examine the violation of the kT factorization approximation for quark production in high energy proton-nucleus collisions. We comment on its implications for the open charm and quarkonium production in collider experiments.
We apply the effective theory of color glass condensate to the analysis of gluon states in dense quark matter, in which the saturation region of gluons is also present. We find that in the region two point function of gluons shows algebraic long range order. The order is completely the same as the one gluons show in the dense quark matter, which form quantum Hall states. The order leads to the vanishing of massless gluon pole. We also find that the saturation region of gluons extends from small $x$ to even large $x\lesssim 1$ in much dense quark matter. We point out a universality that the col…
We modify the McLerran-Venugopalan model to include only a finite number of sources of color charge. We argue that Coulombic interactions between these color charges generates a source-source correlation function that properly includes the effects of color charge screening, a generalization of Debye screening for the Color Glass Condensate. Such a model may be useful for computing angular harmonics of flow measured in high energy hadron collisions for small systems. In this paper we provide a basic formulation of the problem on a lattice.
Noting that the number of gluons in the hadron wave function is discrete, and their formation in the chain of small-x evolution occurs over discrete rapidity intervals of Delta y~1/alpha_s, we formulate the discrete version of the Balitsky--Kovchegov evolution equation and show that its solution behaves chaotically in the phenomenologically interesting kinematic region.
I review the physical and mathematical foundations for the theoretical description of the hadron wavefunction at small $x$ as a Colour Glass Condensate. In this context, I discuss the phenomenon of gluon saturation and some of its remarkable consequences: a new ``geometric scaling'' for $F_2$, which has been recently identified at HERA, and the unitarization of the hadronic cross-sections at high energy. I show that by combining saturation and confinement one obtains cross-sections which saturate the Froissart bound.
We present an evolution equation for the Bjorken x dependence of diffractive dissociation on hadrons and nuclei at high energies. We extend the formulation of Kovchegov and Levin by relaxing the factorization assumption used there. The formulation is based on a technique used by Weigert to describe interjet energy flow. The method can be naturally extended to other exclusive observables.
We show that an observable fraction of the measured elliptic flow may originate in classical gluon fields at the initial stage of a peripheral high-energy nuclear collision. This mechanism complements the contribution of late stage mechanisms, such as those described by hydrodynamics, to the observed elliptic flow.
We extract the leading twist-2 and subleading twist-3 Distribution Amplitudes (DAs) of the rho meson using the HERA data on diffractive rho photoproduction. We do so using several Colour Glass Condensate (CGC) inspired and a Regge inspired dipole models. We find that our extracted twist-2 DA is not much model dependent and is consistent with QCD Sum Rules and lattice predictions. The extracted twist-3 DA is more model dependent but is still consistent with the Sum Rules prediction.
Within the framework of the Color Glass Condensate model, we evaluate quark and gluon Generalized Parton Distributions (GPDs) and the cross section of Deeply Virtual Compton Scattering (DVCS) in the small-$x_{B}$ region. We demonstrate that the DVCS cross section becomes independent of energy in the limit of very small $x_{B}$, which clearly indicates saturation of the DVCS cross section. Our predictions for the GPDs and the DVCS cross section at high-energies can be tested at the future Electron-Ion Collider and in ultra-peripheral nucleus-nucleus collisions at the LHC.
In this work the rapidity and transverse momentum dependence of the nuclear modification ratio is evaluated for the dilepton production at RHIC and LHC, using the Color Glass Condensate (CGC) framework. The ratio is compared for two distinct dilepton mass values, concerning the transverse momentum distribution, and a suppression of the Cronin peak is verified even for large mass. The nuclear modification ratio suppression in the dilepton rapidity spectra, as obtained for hadrons at RHIC, is verified for LHC energies at large transverse momentum, although not present at RHIC energies. These res…
When probed at very high energies or small Bjorken x_bj, QCD degrees of freedom manifest themselves as a medium of dense gluon matter called the Color Glass Condensate. Its key property is the presence of a density induced correlation length or inverse saturation scale R_s=1/Q_s. Energy dependence of observables in this regime is calculable through evolution equations, the JIMWLK equations, and characterized by scaling behavior in terms of Q_s. These evolution equations share strong parallels with specific counterparts in jet physics. Experimental relevance ranges from lepton proton and lepton…
The study of the perturbative Odderon at high gluon densities in the color glass condensate requires to take into account states with more than three gluons. We formulate evolution equations for these states in which the number of gluons is not fixed during the evolution. We further determine the coupling of these Odderon states to the gamma to eta_c impact factor for arbitrary numbers of gluons. We find an exact solution of the evolution equation for the four-gluon Odderon state in terms of the three-gluon Odderon state. It is shown that a similar solution exists also for a different class of…
An exact analytical solution to the Balitsky-Kovchegov equation is proposed incorporating Mellin transformation with saddle point approximation. The theoretical significance of this solution is its simplicity in form which requires very less number of parameters. Using this solution along with the idea of color glass condensate, a very good description of RHIC and LHC data on differential yield is obtained for d-Au, p-p and p-Pb collisions. Quantitative prediction for the nuclear modification factors for p-Pb, Pb-Pb and Xe-Xe collisions relevant for very recent LHC run 2 is yielded.
I give a brief overview of the effective theory for the Color Glass Condensate, which is the high-density gluonic matter which controls high-energy scattering in QCD in the vicinity of the unitarity limit. I concentrate on fundamental phenomena, like gluon saturation, unitarization, and geometric scaling, and the way how these are encoded in the formalism. I emphasize the importance of the next-to-leading order corrections, especially the running of the coupling, for both conceptual and phenomenological issues. I survey the implications of the CGC theory for the HERA physics and its phenomenol…
I consider production of heavy quarks in pA and AA collisions in the framework of the Color Glass Condensate. I discuss the heavy quark production in pA collisions in a classical approximation which takes into account all multiple rescatterings of a proton in a nucleus. For peripheral collisions, heavy quark production cross section can be written in $k_T$-factorized form. The $k_T$ factorization is used to construct a simple model which takes into account both classical and quantum effects in pA and AA collisions. I review the main result of calculation based on that model: open charm in the …
We consider, for the first time, correlations between produced quarks in p-A collisions in the framework of the Color Glass Condensate. We find a quark-quark ridge that shows a dip at $Δη\sim 2$ relative to the gluon-gluon ridge. The origin of this dip is the short range (in rapidity) Pauli blocking experienced by quarks in the wave function of the incoming projectile. We observe that these correlations, present in the initial state, survive the scattering process. We suggest that this effect may be observable in open charm-open charm correlations at the Large Hadron Collider.
We compute the inclusive cross-section of $f_{2}$ tensor mesons production in proton-proton collisions at high-energy. We use an effective theory inspired from the tensor meson dominance hypothesis that couples gluons to $f_{2}$ mesons. We compute the differential cross-section in the $k_{\perp}$-factorization and in the Color Glass Condensate formalism in the low density regime. We show that the two formalisms are equivalent for this specific observable. Finally, we study the phenomenology of $f_{2}$ mesons by comparing theoretical predictions of different parameterizations of the unintegrate…
We complete the construction of the renormalization group equation (RGE) for the Color Glass Condenstate begun in Paper I. This is the equation which governs the evolution with rapidity of the statistical weight function for the color glass field. The coefficients in this equation --- one-loop real and virtual contributions --- are computed explicitly, to all orders in the color glass field. The resulting RGE can be interpreted as the imaginary-time evolution equation, with rapidity as the ``imaginary time'', for a quantum field theory in two spatial dimensions. In the weak field limit it redu…
The similarities between string percolation and Glasma results are emphasized, special attention being paid to rapidity long range correlations, ridge structure and elliptic flow. As the string density of high multiplicity pp collisions at LHC energies has similar value as the corresponding to Au-Au semi-central collisions at RHIC we also expect in pp collisions long rapidity correlations and ridge structure, extended more than 8 units in rapidity.
We calculate the transverse correlation of fluctuations of the deposited energy density in nuclear collisions in the framework of the Gaussian color glass condensate model.
We discover new vacuum solutions of the JIMWLK equation, which correspond to center of a gauge group. We improve the color glass condensate (CGC) model by an explicit usage of a density matrix. Studying scattering of CGC states in an external color field, we observe that an amplitude is naturally expressed via group characters. We construct an example that shows how new thin effects may be potentially observed in peripheral collisions. We prove that at any parton density a gluonic CGC state does not become a true black disk. We find a wave function of a true black disk and show that it necessa…
The rapidity and transverse momentum dependence of the nuclear modification ratio for dilepton production at RHIC and LHC is presented, calculated in the Color Glass Condensate (CGC) framework. The transverse momentum ratio is compared for two distinct dilepton mass values and a suppression of the Cronin peak is verified even for large mass. The nuclear modification ratio suppression in the dilepton rapidity spectra, as obtained experimentally for hadrons at RHIC, is verified for LHC energies at large transverse momentum, although not present at RHIC energies. The ratio between LHC and RHIC nu…
In the recent experiments like DIS at HERA or the heavy-ion experiments at RHIC, and also in expected LHC at CERN, the number of involved partons is very large, due to the high energy and/or the high number of participants of those experiments. These high parton densities should in principal lead to an extremely huge multiparticle production, but experimentally we have seen that this is not the case. So there should be a mechanism that reduces the number of created particle. Here, I review the problem of parton saturation and its implications through three in principal different approaches, bu…
In this review, I present the description of the early stages of heavy ion collisions at high energy in the Color Glass Condensate framework, from the pre-collision high energy nuclear wavefunction to the point where hydrodynamics may start becoming applicable.
We consider the influence of parton saturation in the Color Glass Condensate on the back-to-back azimuthal correlations of high $p_T$ hadrons in $pA$ (or $dA$) collisions. When both near--side and away--side hadrons are detected at mid-rapidity at RHIC energy, the effects of parton saturation are constrained to transverse momenta below the saturation scale $p_T \leq Q_s$; in this case the back-to-back correlations do not disappear but exhibit broadening. However when near-side and away-side hadrons are separated by several units of rapidity, quantum evolution effects lead to the depletion of b…
Recently Kovner and Lublinsky proposed a set of equations which can be viewed as dual to JIMWLK evolution. We show that these dual equations have a natural dipole-like structure, as conjectured by Kovner and Lublinsky. In the high energy large N_c limit these evolution equations reduce to equations previously derived in the dipole model. We also show that the dual evolution kernel is scheme dependent, although its action on the weight functional describing a high energy state gives a unique result.
