In this contribution we briefly review the current status of the dipole models and parton saturation on the basis of results presented at the HERA-LHC workshops in the years 2006-2008. The problem of foundations of the dipole models is addressed within the QCD formalism. Some limitations of the models and open problems are pointed out. Furthermore, we review and compare the currently used dipole models and summarise the applications to describe various sets of HERA data. Finally we outline some of the theoretical approaches to the problem of multiple scattering and saturation.
Partons are defined as the quanta in a Fock space description of a hadron. Gluon saturation is described in the Weizsäcker-Williams approximation for a large nucleus. The elements of DGLAP and BFKL evolution are given with the BFKL equation derived in a large-$N_c$ dipole formalism. A more general discussion of saturation is given in terms of a dipole scattering on a nucleon or nucleus. Possible evidence for saturation at HERA is discussed.
The phenomenology of gluon saturation at small parton momentum fraction, Bjorken-x, in the proton and in the nucleus is introduced. The experimentally-accessible kinematic domains at the nucleus-nucleus colliders RHIC and LHC are discussed. Finally, the saturation hints emerging from measurements at RHIC and the perspectives for LHC are described.
We introduce a generalized definition of parton distribution functions (PDFs) for a more consistent all-order treatment of power corrections. We present a new set of modified DGLAP evolution equations for nuclear PDFs, and show that the resummed $α_s A^{1/3}/Q^2$-type of leading nuclear size enhanced power corrections significantly slow down the growth of gluon density at small-$x$. We discuss the relation between the calculated power corrections and the saturation phenomena.
The idea of partons and the utility of using light-cone gauge in QCD are introduced. Saturation of quark and gluon distributions are discussed using simple models and in a more general context. The Golec-Biernat W\usthoff model and some simple phenomenology are described. A simple, but realistic, equation for unitary, the Kovchegov equation, is discussed, and an elementary derivation of the JIMWLK equation is given.
This is a personal summary of the meeting "Saturation, the Color Glass Condensate and Glasma: What Have we Learned from RHIC?" that took place at BNL in May 2010. The purpose of the meeting was to discuss the status of high density QCD and parton saturation, and to review the progress that RHIC has allowed in the field.
It is argued that the gluon density introduced by A.Mueller in relation to the saturation phenomenon is different from the standardly defined one, which turned out to have a soliton-like form in the numerical calculations. It is shown that A.Mueller's densities obtained from the latter possess the same saturation properties as originally predicted.
Quark and gluon distributions in the light-cone wavefunction of a high energy hadron or nucleus are calculated in the saturation regime. One loop calculations are performed explicitly using the equivalence between the parton distribution in the light-cone wavefunction and the production distribution of that parton in a current-nucleon (nucleus) scattering. We argue that, except for some overall numerical factors, the Weizsacker- Williams wavefunction correctly gives the physics of the gluon distribution in a light-cone wavefunction.
We report on the results obtained in the study of the parton saturation effects, taken into account through the multiscattering Glauber-Mueller approach, applied to the Drell-Yan (DY) process described in the color dipole picture. As a main result, one shows that those effects play an important role in the estimates of the DY differential cross section atRHIC energies.
We construct diffractive parton distributions in the model of DIS diffraction in which the diffractive state is formed by the $q\bar{q}$ and $q\bar{q}g$ components. The interaction of such systems with the proton is described by the dipole cross section given by the saturation model. We find Regge factorization property with the measured at HERA depenedence on energy. The found parton distributions are evolved with the DGLAP evolution equations to make a comparison with the data.
We derive the medium-induced, coherent gluon radiation spectrum associated with the hard forward scattering of an energetic parton off a nucleus, in the saturation formalism and within the Gaussian approximation for the relevant correlators of Wilson lines. The calculation reproduces the simple expression for the spectrum previously obtained in the opacity expansion formalism, and rigorously specifies its validity range. The connection between the calculations in the opacity expansion and saturation formalisms is made apparent. This study may serve as a first step in order to implement consist…
We present an implementation of a parton shower algorithm for hadron colliders and electron-positron colliders based on the dipole factorisation formulae. The algorithm treats initial-state partons on equal footing with final-state partons. We implemented the algorithm for massless and massive partons.
We derive two coupled non-linear evolution equations corresponding to the truncation of the Balitsky infinite hierarchy of saturation equations after inclusion of dipole-dipole correlations, i.e. one step beyond the Balitsky-Kovchegov (BK) equation. We exhibit an exact solution for maximal correlation which still satisfies the same asymptotic geometric scaling as BK but with the S-matrix going to 1/2 (instead of 0) in the full saturation region.
The production of $c\bar{c}c\bar{c}$, $b\bar{b}b\bar{b}$ and $c\bar{c}b\bar{b}$ pairs considering double parton scattering at LHC energies is investigated. We estimate the contribution of saturation effects to the different final states and predict the energy dependence of the cross sections. Moreover, we estimate the ratio between the double and single parton scattering cross sections for the full rapidity range of the LHC and for the rapidity range of the LHCb experiment. For the full rapidity range we confirm a previous prediction, namely that for charm production the double parton scatteri…
The quark saturation behavior at low $Q^2$ is shown in a numeric solution of the DGLAP equation with parton recombination corrections, which resembles the widely discussed JIMWLK saturation of gluons. Our calculation suggests that the partonic saturation can be interpreted as a dynamical balance between the splitting and the fusion processes of partons, without any other condensation mechanisms added. The nuclear shadowing saturation at small $x$ resulted from the proposed quark saturation is also discussed.
We review diffractive deep inelastic scattering (DIS) in the light of the collinear factorization theorem. This theorem allows to define diffractive parton distributions in the leading twist approach. Due to its selective final states, diffractive DIS offers interesting insight into the form of the diffractive parton distributions which we explore with the help of the saturation model. We find Regge-like factorization with the correct energy dependence measured at HERA. A remarkable feature of diffractive DIS is the dominance of the twist-4 contribution for small diffractive masses. We quantif…
We note that the phenomenon of perturbative saturation leads to transverse momentum broadening in the spectrum of partons produced in hadronic collisions. This broadening has a simple interpretation as parton level Cronin effect for systems in which saturation is generated by the "tree level" Glauber-Mueller mechanism. For systems where the broadening results form the nonlinear QCD evolution to high energy, the presence or absence of Cronin effect depends crucially on the quantitative behavior of the gluon distribution functions at transverse momenta kt outside the so called scaling window. We…
We discuss the description of the proton structure function within the dipole factorisation framework. We parametrise the forward dipole amplitude to account for saturation as predicted by the small-x QCD evolution equations. Contrarily to previous models, the saturation scale does not decrease when taking heavy quarks into account. We show that the same dipole amplitude also allows to reproduce diffractive data and exclusive vector meson production.
These lecture notes are a combination of two lectures and a contribution to the celebration of Andrzej Białas' birthday at the LI Cracow School of Theoretical Physics in June 2011. I here discuss the dynamics of particle production in high energy reactions. It includes parton cascades and hadronization in e+e--ann., small x evolution including the Double Leading Log approximation and the BFKL equation, saturation at high densities and the BK equation, and finally the Lund Dipole Cascade model for high energy collisions, which is implemented in the DIPSY MC.
At high center of mass energies, hadroproduction of heavy quarks can be expressed in terms of the same color dipole cross section as low Bjorken-x deep inelastic scattering. We show analytically that at leading order, the dipole formulation is equivalent to the gluon-gluon fusion mechanism of the conventional parton model. In phenomenological application, we employ a parameterization of the dipole cross section which also includes higher order and saturation effects, thereby going beyond the parton model. Numerical calculations in the dipole approach agree well with experimental data on open c…
In this publication the implementation of a new parton shower model based on the Catani-Seymour dipole factorisation, as first suggested by Z. Nagy and D. E. Soper, is discussed. First results obtained with the new algorithm are compared with experimental data.
In this paper we calculate dipole amplitude and dipole amplitude correlations using first and second equation from Balitsky-Kovchegov hierarchy. Our analysis shows that even in presence of weak dipole correlation in initial condition mean field approximation breaks down through evolution. This difference asymptotically grows not only by absolute value but also by relative difference of dipole correlation from mean field value. This could affect physical values related to dipole correlation such as elliptic flow $v_2$ and 'back-forward' asymmetry in saturation model.
In this paper we study the description of saturation in Balitsky, Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and Kovner (B-JIMWLK) picture when restricted to observables made up only from dipole operators. We derive a functional form of the evolution equation for the dipole probability distribution and find a one-parameter family of exact solutions to the dipole evolution equations.
Parton distributions in the small $x$ region are numerically predicted by using a modified DGLAP equation with the GRV-like input distributions. We find that gluon recombination at twist-4 level obviously suppresses the rapid growth of parton densities with $x$ decrease. We show that before the saturation scale $Q^2_s$ is reached, saturation and partial saturation appear in the small $x$ behavior of parton distributions in nucleus and free proton, respectively. The antishadowing contributions to the saturation phenomena are also discussed.
A modification of the collinear evolution equations as an appropriate approach to improve the behavior of parton distribution functions in the region of small longitudinal momentum fractions, and to find more theoretical arguments to clarify the possible appearence of saturation regime is suggested. It is argued that parton diffusion in the rapidity space at large parton densities along the space-time evolution could result in the emergence of a natural saturation scale on which freezing actually occurs.
We use data on the deep inelastic structure function F_2 in order to constrain the cross-section for scattering a colour dipole off a proton. The data seem to prefer parameterisations which include saturation effects. That is they indicate that the strong rise with energy of the dipole cross-section, which holds for small dipoles, pertains only for r < r_s(x) where r_s(x) decreases monotonically as x decreases. Subsequent predicitions for the diffractive structure function F_2^{D(3)} also hint at saturation, although the data are not really sufficiently accurate.
A parton-shower model for hadronic collisions based on the emission properties of QCD dipoles is proposed. This proposal therefore extends the well-known radiation pattern of pure final-state colour dipoles to QCD initial-state radiation, both of which are treated perturbatively. Corresponding dipole splitting functions are derived and the kinematics of all dipole splittings is discussed. Application to hadron production in electron-positron annihilation, to Drell-Yan lepton-pair and QCD jet production yields encouraging results.
In the framework of a toy model which possesses the main features of QCD in the high energy limit, we conduct a numerical study of scattering amplitudes constructed from parton splittings and projectile-target multiple interactions, in a way that unitarizes the amplitudes without however explicit saturation in the wavefunction of the incoming states. This calculation is performed in two different ways. One of these formulations, the closest to field theory, involves the numerical resummation of a factorially divergent series, for which we develop appropriate numerical tools. We accurately comp…
This talk reviews briefly some of the main results of the small-x dipole formulation with regards to unitarity corrections. It illustrates the correspondence between unitarity and saturation corrections in the dipole approach and multiple t-channel pomeron exchange in the traditional BFKL view, and discusses how one can estimate and understand the effects of saturation.
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…
We present a new parton-shower algorithm. Borrowing from the basic ideas of dipole cascades, the evolution variable is judiciously chosen as the transverse momentum in the soft limit. This leads to a very simple analytic structure of the evolution. A weighting algorithm is implemented, that allows to consistently treat potentially negative values of the splitting functions and the parton distributions. We provide two independent, publicly available implementations for the two event generators Pythia and Sherpa.
We present the numerical solution of the non-linear evolution equation for DIS on nuclei for $x = 10^{-2} ÷10^{-7}$. We demonstrate that the solution to the non-linear evolution equation is quite different from the Glauber - Mueller formula which was used as the initial condition for the equation. We illustrate the energy profit for performing DIS experiments on nuclei. However, it turns out that the gain is quite modest: $x_{Au} \simeq 5 x_{\rm proton} $ for the same parton density. We find that the saturation scale $Q^2_s \propto A^{1/3}$. For gold the saturation scale $Q_{s,Au} \simeq 1.5 G…
We pose the question: how far can we push the QCD dipole model and the idea of QCD saturation to the kinematical regions traditionally associated with soft nonperturbative physics? The answer is that it works sufficiently well allowing a smooth transition from hard (BFKL) pomeron to the soft one.
We employ values of the colour dipole cross section extracted from electroproduction and photoproduction data to show that gluon saturation effects are not required by the current HERA data but will become important in the THERA energy region.
We propose a new criterium for saturation of the density of partons both in nucleons and nuclei. It is applicable to any multiple scattering model which would be used to compute the number of strings exchanged in $ep$ and $eA$ collisions. The criterium is based on percolation of strings, and the onset of percolation is estimated from expectations coming from the study of heavy ion collisions at high energies. We interpret this onset as an indication of saturation of the density of partons in the wave function of the hadron. In order to produce quantitative results, a particular model fitted to…
Parton evolution with the rapidity essentially is a branching diffusion process. We describe the fluctuations of the density of partons which affect the properties of QCD scattering amplitudes at moderately high energies. We arrive at different functional forms of the latter in the case of dipole-nucleus and dipole-dipole scattering.
Evolution equations for multiplicities in QCD cascades can, both in the parton and dipole picture, be used to estimate corrections beyond the formal accuracy of the modified leading log approximation (MLLA). The differences between the two pictures, and other uncertainties beyond first order MLLA corrections, are here investigated in some detail. For example, I discuss how some colour suppressed corrections, which cannot be determined without better knowledge of non-perturbative QCD, are related to the question of colour reconnection. A generalized evolution equation for the dipole cascade is …
In this paper we analyse the exclusive vector meson photoproduction in the recent run using Xenon-ions at energy of 5.44 TeV performed by the Large Hadron Collider. We focus on the ultraperipheral collisions and provide theoretical predictions for coherent and incoherent cross sections within the color dipole approach and gluon saturation framework. The rapidity distribution is investigated in both cases and comparison to other approaches available at literature is done. We show that the expected yields are enough to perform reliable cross section measurements for light mesons as $ρ^0$ and $ϕ$…
We argue that the suppression of high p_t hadrons discovered recently in heavy ion collisions at RHIC may be a consequence of saturation in the Color Glass Condensate. We qualitatively and semi-quantitatively describe the data, in particular, the dependence upon the number of nucleon participants. We show that if parton saturation sets in at sufficiently small energy, then in nucleus-nucleus collisions at RHIC and LHC energies the cross sections of semi-hard processes should scale approximately with the number of participants, N_{part}. Our results provide a possible explanation of both the ab…
We evaluate the unintegrated gluon distribution of the proton starting from a parametrization of the color dipole cross section including DGLAP evolution and saturation effects. To this end, we perform the Fourier-Bessel transform of $σ(x,r)/α(r)$. At large transverse momentum of gluons we match the so-obtained distribution to the logarithmic derivative of the collinear gluon distribution. We check our approach by calculating the proton structure function $F_L(x,Q^2)$ finding good agreement with HERA data.
We develop a dipole model for HERA DIS data which incorporates the impact parameter distribution of the proton. The model describes the inclusive total $γ^*p$ cross-sections as well as the diffractive $J/ψ$ differential cross-sections. We compare the model with previous approaches and show that the $t$-distributions are sensitive to saturation phenomena. We estimate the boundary of the saturation region and show that it dominates the data in the low-$Q^2$ region where the total $γ^*p$ cross-section exhibits the same universal rise as hadronic cross-sections. The model is then extended to nucle…
I describe the parton picture at strong coupling emerging from the gauge/gravity duality, with emphasis on the universality of the phenomenon of parton saturation. I discuss several consequences of this picture for the phenomenology of a strongly coupled quark-gluon plasma, which are potentially relevant for heavy ion collisions at RHIC and LHC.
We review the recent finding of the two-plateau momentum distribution of sea quarks in deep inelastic scattering off nuclei in the saturation regime. The diffractive plateau which dominates for small $\bp$ measures precisely the momentum distribution of quarks in the beam photon, the rôle of the nucleus is simply to provide an opacity. The plateau for truly inelastic DIS exhibits a substantial nuclear broadening of the momentum distribution. Despite this nuclear broadening, the observed final state and initial state sea quark densities do coincide exactly. We emphasize how the saturated sea is…
We review QCD based descriptions of diffractive deep inelastic scattering emphasizing the role of models with parton saturation. These models provide natural explanation of such experimentally observed facts as the constant ratio of diffractive and total DIS cross sections as a function of the Bjorken variable x and Regge factorization of diffractive parton distributions.
The idea of saturation of parton densities in small x physics is briefly introduced. Some aspects of saturation are described, mainly focusing on the status of our knowledge on the non-linear equations describing the high parton density regime. Implications of saturation ideas on the description of nuclear collisions at the Relativistic Heavy Ion Collider are discussed.
In this paper we demonstrate that the dense system of partons (gluons) can be produced in dilute-dilute system scattering, using the example of dipole-dipole collisions. This increase in density stems from the intensive gluon cascades that can be described by the enhanced BFKL Pomeron diagrams (Pomeron loops). For the first time we found the analytical solution to the equation for diffraction production in the dipole-dense parton system scattering, using the simplified BFKL kernel. Having this solution as well as the solution to Balitsky- Kovchegov equation we developed technique that allowed …
We investigate $ρ$ vector meson production in $ep$ collisions at HERA with leading neutrons in the dipole formalism. The interaction of the dipole and the pion is described in a mixed-space approach, in which the dipole-pion scattering amplitude is given by the Marquet-Peschanski-Soyez saturation model, which is based on the travelling wave solutions of the nonlinear Balitsky-Kovchegov equation. We estimate the magnitude of the absorption effects and compare our results with a previous analysis of the same process in full coordinate space. In contrast with this approach, the present study lead…
We show that the geometric scaling of the dipole cross section can be explained using standard DGLAP perturbative evolution. The DGLAP improved saturation model due to the Laplace transforms method is considered at LO and NNLO approximations from the experimental data by relying on a Froissart-bounded parametrization of $F_{2}(x,Q^{2})$. These results are comparable with Golec-Biernat-W$\ddot{\mathrm{u}}$sthoff (GBW) model in a wide kinematic region $rQ_{s}$ which takes into account charm mass. The successful description of $σ_{\mathrm{dip}}(x,r)/σ_{0}$ and $σ_{\mathrm{dip}}(rQ_{s})/σ_{0}$ are…
In the kinematical region where the center of mass energy is much larger than all other scales, the Drell-Yan process can be formulated in the target rest frame in terms of the same color dipole cross section as low Bjorken-x deep inelastic scattering. Since the mechanisms for heavy dilepton production appear very different in the dipole approach and in the conventional parton model, one may wonder whether these two formulations really represent the same physics. We perform a comparison of numerical calculations in the color dipole approach with calculations in the next-to-leading order parton…
This paper studies two related questions in high energy onium-onium scattering: the probability of producing an unusually large number of particles in a collision, where it is found that the cross section for producing a central multiplicity proportional to $k$ should decrease exponentially in $\sqrt{k}$. Secondly, the nature of gluon (dipole) evolution when dipole densities become so high that saturation effects due to dipole-dipole interactions become important: measures of saturation are developed to help understand when saturation becomes important, and further information is obtained by e…
We discuss the recent claim that hadron multiplicities measured at RHIC energies are directly described in terms of gluon degrees of freedom fixed from the initial conditions of central heavy ion collisions. The argument is based on the parton saturation scenario expected to be valid at high parton densities and on the assumption of conserved gluon number. Alternatively we conjecture that "bottom-up" equilibration before hadronization modifies this picture, due to nonconservation of the number of gluons.
We derive the two-plateau momentum distribution of final state (FS) quarks produced in deep inelastic scattering (DIS) off nuclei in the saturation regime. The diffractive plateau which dominates for small transverse momentum p measures precisely the momentum distribution of quarks in the beam photon, the role of the nucleus is simply to provide an opacity. The plateau for truly inelastic DIS exhibits a substantial nuclear broadening of the FS momentum distribution. We discuss the relationship between the FS quark densities and the properly defined initial state (IS) nuclear quark densities.Th…
Here, we discuss QCD predictions on multiplicities in parton and dipole approaches. The most general treatment is based on the notion of the generating functions. The generating function G is defined as $G(u,y)=\sum_nu^nP_n(y)$, where $P_n$ is the probability of the n-particle production at energy denoted by y, u is an auxiliary variable. The mean multiplicity and higher moments of the multiplicity distribution $P_n$ are given by the u-derivatives of G at u=1.
Dihadron angular correlations in forward $pA$ collisions have been considered as one of the most sensitive observables to the gluon saturation effects. In general, both parton shower effects and saturation effects are responsible for the back-to-back dihadron angular de-correlations. With the recent progress in the saturation formalism, we can incorporate the parton shower effect by adding the corresponding Sudakov factor in the saturation framework. In this paper, we carry out the first detailed numerical study in this regard, and find a very good agreement with previous RHIC $pp$ and $dAu$ d…
The dipole subtraction method for calculating next-to-leading order corrections in QCD was originally only formulated for massless partons. In this paper we extend its definition to include massive partons, namely quarks, squarks and gluinos. We pay particular attention to the quasi-collinear region, which gives rise to terms that are enhanced by logarithms of the parton masses, M. By ensuring that our subtraction cross section matches the exact real cross section in all quasi-collinear regions we achieve uniform convergence both for hard scales Q ~= M and Q >> M. Moreover, taking the masses t…
We study the energy dependence of the saturation momentum in the context of the collinearly improved Leading and Next to Leading BFKL evolution, and in the presence of saturation boundaries. We find that the logarithmic derivative of the saturation momentum is varying very slowly with Bjorken-x, and its value is in agreement with the Golec-Biernat and Wusthoff model in the relevant x region. The scaling form of the amplitude for dipole-dipole or dipole-hadron scattering in the perturbative side of the boundary is given.
