Gluon Saturation and the Formation Stage of Heavy Ion Collisions
Larry McLerran
备注：
Paper prepared for an edition of the Landolt-Bernstein series: Numerical Data and Functional Relationships, Editor in Chief, W. Martienssen, Volume 1/23, Relativistic Heavy Ion Physics, edited by R. Stock
The high energy limit of QCD is controlled by very high energy density gluonic matter, the Color Glass Condensate. In the first instants of the collisions of two sheets of Colored Glass Condensate, a Glasma is formed with longitudinal flux tubes of color electric and magnetic fields. These flux tubes decay and might form a turbulent liquid that eventually thermalizes into a Quark Gluon Plasma
A new geometrical scaling method with gluon saturation rapidity limit is proposed to study the gluon saturation feature of the central rapidity region of relativistic nuclear collisions. The net-baryon number is essentially transported by valence quarks that probe the saturation regime in the target by multiple scattering. We take advantage of the gluon saturation model with geometric scaling of the rapidity limit to investigate the net baryon distributions, nuclear stopping power and gluon saturation features in the SPS and RHIC energy regions. Predictions for net-baryon rapidity distribution…
Progresses towards the calculation and the understanding of NLO/NLL contributions to Deep Inelastic Scattering at low x with gluon saturation are being reviewed.
The leading order single inclusive soft gluon production in high energy proton-nucleus (pA) collisions has been studied by various approaches for more than two decades. The first correction due to the gluon saturation in proton was analytically attempted recently through a diagrammatic approach in which only partial results were obtained. An important feature of the first saturation correction is that it includes both initial state and final state interactions. In this paper, we present the complete result derived from the Color Glass Condensate framework. Our approach is to analytically solve…
Deep inelastic scattering at low x can be described by essentially only two fitted parameters. The interpretation of J/Psi photoproduction in terms of the gluon structure function is elaborated upon.
We consider a novel mechanism for J/Psi production in nuclear collisions arising due to the high density of gluons. We calculate the resulting J/Psi production cross section as a function of rapidity and centrality. We evaluate the nuclear modification factor and show that the rapidity distribution of the produced J/Psi's is significantly more narrow in AA collisions due to the gluon saturation effects. Our results indicate that gluon saturation in the colliding nuclei is a significant source of J/Psi suppression that can be disentangled from the quark-gluon plasma effects.
We calculate the unintegrated gluon distribution at low intrinsic transverse momenta and its parameters are found from the best description of the SPS and LHC data on the $pp$ collision in the soft kinematical region. It allows us to study the saturation of the gluon density at low $Q^2$ more carefully and find the saturation scale.
We report on the interesting possibility of instanton-driven gluon saturation in lepton-nucleon scattering at small Bjorken-x. The explicitly known instanton gauge field serves as a concrete realization of an underlying non-perturbative saturation mechanism associated with strong classical fields. The conspicuous, intrinsic instanton size scale known from lattice simulations, turns out to determine the saturation scale. The ``colour glass condensate'' can be identified in our approach with the QCD-sphaleron state, dominating instanton-induced processes in the softer regime.
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.
We present an overview of theoretical aspects of the phenomenon of gluon saturation in high energy scattering in Quantum Chromo Dynamics. Then we review the state-of-the-art of saturation-based phenomenological approaches to the study and characterisation of the initial state of ultra-relativistic heavy ion collisions performed at RHIC and the LHC. Our review focuses mostly in the Color Glass Condensate effective theory, although we shall also discuss other approaches in parallel.
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 show that the inclusion of parton density effects in the perturbative small-x evolution reduces the strength of the powerlike growth of total hadronic cross sections.
Our approach to low-x deep inelastic scattering based on the ell-perp factorization of perturbative QCD (the color-dipole picture) yields parameter-free absolute predictions for J/Psi production. The connection of J/Psi production to the low-x saturation scale and to the gluon structure function is clarified.
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…
Gluon distribution function at small x is computed using the extended McLerran-Venugopalan model. A lipatov like enhancement is seen as well as a non-trivial transverse momentum dependence which is estimated. It is shown that gluon density saturates at small x which would lead to unitarization of cross sections computed with this distribution function.
The CCFM equation and its extended form with a quadratic term (KGBJS equation) are solved with fixed and running coupling constant. The solution of the KGBJS equation is compared to gluon densities resulting from the CCFM and BK equations. As the saturation scale $Q_s$ now becomes available as a function of the hard scale $p$ we observe that low values of $p$ impede its growth with $\frac 1 x$. Also, at values much larger than partons transversal momentum the saturation effects become independent on the hard scale what we call liberation of saturation scale. We also introduce the hard-scale-re…
We construct a Gaussian approximation to the effective theory for the Colour Glass Condensate which describes correctly the gluon distribution both in the low density regime at high transverse momenta (above the saturation scale $Q_s$), and in the high density regime below $Q_s$, and provides a simple interpolation between these two regimes. At high momenta, the effective theory reproduces the BFKL dynamics, while at low momenta, it exhibits gluon saturation and, related to it, colour neutrality over the short distance scale $1/Q_s \ll 1/Λ_{QCD}$. Gauge--invariant quantities computed within th…
At very high energies, the relevant component of the hadron wavefunction can be described as a Color Glass Condensate, i.e., a state of high density gluonic matter whose distribution is random, but frozen over the relevant time scales. The weight function for this distribution obeys a renormalization group equation in the form of a functional Fokker-Planck equation. Its solution leads to an effective theory which predicts gluon saturation at sufficiently high energy, or small Bjorken's $x$.
Standard perturbative calculations lead to pathologically large NLO corrections to low-$x_{Bj}$ evolution equations like BFKL and BK. Using a more refined treatment of kinematics in mixed-space, relevant when gluon saturation sets on, one obtains an improved version of the BK equation, resumming to all orders the most severe of those large higher order corrections.
At very high energies we expect that the hadronic cross sections satisfy the Froissart bound, which is a well-established property of the strong interactions. In this energy regime we also expect the formation of the Color Glass Condensate, characterized by gluon saturation and a typical momentum scale: the saturation scale $Q_s$. In this paper we show that if a saturation window exists between the nonperturbative and perturbative regimes of Quantum Chromodynamics (QCD), the total cross sections satisfy the Froissart bound. Furthermore, we show that our approach allows us to describe the high …
We study the effects of gluon saturation on pion production in high energy proton-nucleus collisions using the color glass condensate model. At high p_perp, we show that the p_perp-distribution of gluons behaves as ~ 1/(p_perp^7) in accordance with both conventional perturbative QCD calculations and experiment. Fragmentation of gluons into pions leads to a rapidity dependent depletion of pions relative to the conventional perturbative QCD predictions. We argue that these clear and systematic differences provide a signal for the onset of gluon saturation which is accessible in upcoming experime…
Almost 40 years ago, Gribov and colleagues at the Leningrad Nuclear Physics Institute developed the ideas that led to the Dokhsitzer-Gribov-Altarelli-Parisi the Baltisky-Fadin-Kuraev-Lipatov equations. These equations describe the evolution of the distributions for quarks and gluon inside a hadron to increased resolution scale of a probe or to smaller values of the fractional momentum of a hadronic constituent. I motivate and discuss the generalization required of these equations needed for high energy processes when the density of constituents is large. This leads to a theory of saturation re…
This is a short presentation of our talks given at eRHIC Workshop at the BNL. We give here a status report of our attempts to understand how gluon saturation will manifest itself in deep inelastic scattering with nuclei. This summary reflects our current understanding and shows directions of our research rather then a final answer to the question. Nevertheless, we are able to share with our reader our tentative answer to the question:``Why do we need to measure DIS with nuclei and why these data will be complementary to the information obtained from proton DIS".
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.
We investigate photo-production of vector mesons J/Psi and Upsilon measured both at HERA and LHC, using 2 particular fits of inclusive unintegrated gluon distributions, based on non-linear Balitsky-Kovchegov evolution (Kutak-Sapeta gluon; KS) and next-to-leading order Balitsky-Fadin-Kuraev-Lipatov evolution (Hentschinski-Sabio Vera-Salas gluon; HSS). We find that linear next-to-leading order evolution can only describe production at highest energies, if perturbative corrections are increased to unnaturally large values; rendering this corrections to a perturbative size, the growth with energy …
We analyze three jet production in the central-forward and forward rapidity regions in proton-proton and proton-lead collisions at LHC energies. Our calculation relies on high-energy factorization with a single off-shell gluon obeying small x evolution equation which includes saturation. The calculations are made using two independent Monte Carlo codes implementing tree-level gauge invariant off-shell matrix elements. We calculate differential cross sections for azimuthal decorrelations and unbalanced jet transverse momenta and discuss them in the context of differences in the evolution of the…
We study the gluon content of a large nucleus (i) in the semi-classical McLerran-Venugopalan model and (ii) in the high energy limit as given by the quantum evolution of the Color Glass Condensate. We give a simple and qualitative description of the Cronin effect and high-pT suppression in proton-nucleus collisions.
In this paper we discuss the example of one rapidity gap process: the inclusive cross sections of the leading neutrons in deep inelastic scattering with protons (DIS). The equations for this process are proposed and solved, giving the example of theoretical calculation of the survival probability for one rapidity gap processes. It turns out that the value of the survival probability is small and it decreases with energy
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…
Tracing Gluon Saturation through Hadronization at EIC
Central China Normal University（华中师范大学）；Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所）；Shandong University（山东大学）；École Polytechnique, Institut Polytechnique de Paris（巴黎理工学院巴黎综合理工学校）
In this paper we extend the color dipole formalism to the study of leading neutron production in $e + p \rightarrow e + n + X$ collisions at high energies and estimate the related observables, which were measured at HERA and may be analysed in future electron-proton ($ep$) colliders. In particular, we calculate the Feynman $x_F$ distribution of leading neutrons, which is expressed in terms of the pion flux and the photon-pion total cross section. In the color dipole formalism, the photon-pion cross section is described in terms of the dipole-pion scattering amplitude, which contains informatio…
We investigate the energy dependence of the photo-production cross-section of vector mesons $J/Ψ$ and $Υ$, measured by both HERA experiments H1 and ZEUS in electron-proton collisions and by LHC experiments ALICE, CMS and LHCb in ultra-peripheral proton-proton and ultra-peripheral proton-lead collisions. Our study uses 2 particular fits of inclusive unintegrated gluon distribution, based on non-linear Balitsky-Kovchegov evolution (Kutak-Sapeta gluon; KS) and next-to-leading order Balitsky-Fadin-Kuraev-Lipatov evolution (Hentschinski-Sabio Vera-Salas gluon; HSS). We find that linear next-to-lead…
We modified the gluon saturation model by rescaling the momentum fraction according to saturation momentum and introduced the Cooper-Frye hydrodynamic evolution to systematically study the pseudo-rapidity distributions of final charged hadrons at different energies and different centralities for Au-Au collisions in relativistic heavy-ion collisions at BNL Relativistic Heavy Ion Collider (RHIC). The features of both gluon saturation and hydrodynamic evolution at different energies and different centralities for Au-Au collisions are investigated in this paper.
Including the new HERA data, the $γ^* p$ total cross section is analysed in the generalized vector dominance/colour-dipole picture (GVD/CDP) that contains scaling in $η= (Q^2 + m^2_0) / Λ^2 (W^2)$, where $Λ^2 (W^2)$ is an increasing function of $W^2$. At any $Q^2$, for $W^2 \to \infty$, the cross sections for virtual and real photons become identical, $σ_{γ^* p} (W^2, Q^2) / σ_{γp} (W^2) \to 1$. The gluon density deduced from the colour-dipole cross section fulfills the leading order DGLAP relationship. Evolution à la DGLAP breaks down for $η\lsim 0.1$.
Two-particle azimuthal angle correlations have been proposed to be one of the most direct and sensitive probes to access the underlying gluon dynamics involved in hard scatterings. In anticipation of an Electron-Ion Collider (EIC), detailed studies of dihadron correlation measurements in electron-proton and electron-ion collisions at an EIC have been performed. The impact of such measurements on the understanding of the different gluon distribution functions, as a clean signature for gluon saturation and to constrain saturation models further, has been explored. It is shown that dihadron corre…
We study properties of high energy factorisable gluon densities focusing on saturation effects. In particular we show that the property of saturation of unintegrated gluon density allows for introduction of thermodynamical entropy associated with production of gluons. This is due to the observation that saturation scale acts like a mass of a gluon which related with temperature via thermodynamical relations allows for calculations of entropy. We also show that obtained entropy behaves like multiplicity of produced gluons.
We have developed a model in which the quantum fluctuations of the proton structure are characterised by hot spots, whose number grows with decreasing Bjorken-$x$. Our model reproduces the $F_2(x,Q^2)$ data from HERA at the relevant scale, as well as the exclusive and dissociative $\mathrm{J/}ψ$ photoproduction data from H1 and ALICE. Our model predicts that for $W_{γ\mathrm{p}} \approx 500$ GeV, the dissociative $\mathrm{J/}ψ$ cross section reaches a maximum and then decreases steeply with energy, which is in qualitatively good agreement to a recent observation that the dissociative $\mathrm{…
The universal traveling wave solution to the Balitsky-Kovchegov equation with running coupling (and other equations in the same universality class) is extended to subleading orders at large rapidity and small dipole size $r$. The large rapidity expansion of the logarithm of the saturation scale $Q_s(Y)$ is derived from that traveling wave solution. In addition to the two already known leading terms in $Y^{1/2}$ and $Y^{1/6}$, which are determined only by the LL BFKL kernel, the three following terms in $Y^0$, $Y^{-1/6}$ and $Y^{-1/3}$ are obtained. They are universal and sensitive to NLL BFKL …
In the previous work hep-ph/0112140, hep-ph/0204277, we showed that while the nonlinear QCD evolution equation of Balitsky and Kovchegov (BK) leads to saturation of the scattering amplitude locally in impact parameter space, it does not unitarize the total cross section. This result was recently challenged by Ferreiro, Iancu, Itakura and McLerran (FIIM) who claim that the dipole-hadron cross section computed from the BK equation saturates the Froissart bound. In this comment, we point to a fundamental error in the argument of FIIM which invalidates their conclusion. We show that if the total c…
We study the production of transversely polarized Lambda hyperons in high-energy collisions of protons with large nuclei. The large gluon density of the target at saturation provides an intrinsic semi-hard scale which should naturally allow for a weak-coupling QCD description of the process in terms of a convolution of the quark distribution of the proton with the elementary quark-nucleus scattering cross section (resummed to all twists) and a fragmentation function. In this case of transversely polarized Lambda production we employ a so-called polarizing fragmentation function, which is an od…
We investigate the scale dependence of transverse momentum dependent(TMD) gluon distribution in saturation regime. We found that in the Collins-2011 scheme, the scale dependence of small x gluon TMD is governed by the same renormalization group(RG) equation that holds at moderate or large $x$. Following the standard procedure, one then can resum both double leading logarithm and single leading logarithm in saturation regime by jointly solving the Collins-Soper equation and the RG equation.
The transverse momentum distribution of soft hadrons and jets that accompany central hard-scattering production at hadron colliders is of great importance, since it has a direct bearing on the ability to separate new physics signals from Standard Model backgrounds. We compare the predictions for the gluonic k_t-distribution using two different approaches: resummed NLL BFKL and DGLAP evolution. We find that as long as the initial and final virtualities (k_t) along the emission chain are not too close to each other, the NLL resummed BFKL results do not differ significantly from those obtained us…
Atomic number ($A$) and energy ($\roots$) scaling exponents of multiplicity and transverse energy in heavy ion collisions are analytically derived in the perturbative QCD + saturation model. The exponents depend on the small-$x$ behaviour of gluon distribution functions at an $x$-dependent scale. The relation between initial state and final state saturation is also discussed.
Saturation of gluon distribution is a consequence of the non-linear evolution equations of QCD. Saturation implies the existence of so called saturation momentum which is defined as a gluon density per unit rapidity per transverse area. At large energies for certain kinematical domains saturation momentum is the only scale for physical processes. As a consequence different observables exhibit geometrical scaling (GS). We discuss a number of examples of GS in different reactions.
We study saturation effects in the production of dijets in p-p and p-Pb collisions using the framework of high energy factorization. We focus on central-forward jet configurations, which allow for probing gluon density at low longitudinal momentum fraction. We find significant suppression of the central-forward jet decorrelations in p-Pb compared to p-p, which we attribute to saturation of gluon density in nuclei.
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…
We show that saturation can manifest itself in central-forward dijet production in p-A collisions. In spite of large transverse momenta of the jets, the almost back-to-back dijet configurations are able to probe gluon density at low x and low kt. We perform our study in the framework of high energy factorization with the unintegrated gluon density given by a nonlinear QCD evolution equation. We show that the formalism can successfully account for features measured in e-p and p-p data and it predicts significant suppression of the central-forward jet decorrelations in p-Pb compared to p-p, whic…
Constraining the Dipole Amplitude at NLO with HERA Data and Probing Gluon Saturation in SIDIS
Department of Physics, University of Jyv\"askyl\"a, P.O. Box 35, 40014 University of Jyv\"askyl\"a, Finland；Helsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki, Finland
We consider forward pion production in p p collisions at RHIC energies,which probes the so-called Extended Geometric Scaling region. Upon inclusion of small-x effects via an anomalous dimension within the Color Glass Condensate formalism at leading order in alpha_s, a good description of the cross section as a function of the transverse momentum of the produced pion is obtained. The latter is essential for extractions of the Sivers effect from polarized p p collisions, since it is a sensitive probe of the slope of the cross section. Hence, the presented approach is well suited to extract the S…
The impact parameter dependent gluon distribution of the proton xG(x,Q^2,b) is investigated in a loop-loop correlation model that respects the S-matrix unitarity condition in impact parameter space. We find low-x saturation of G(x,Q^2,b) as a manifestation of S-matrix unitarity. The integrated gluon distribution xG(x,Q^2) does not saturate because of the growth of the effective proton radius with decreasing x.
Exploiting recently obtained analytic solutions of classical Yang-Mills equations for higher order perturbations in the field of the dilute object (proton), we derive the complete first saturation correction to the single inclusive semi-hard gluon production in high energy proton-nucleus collisions by applying the Lehmann-Symanzik-Zimmermann reduction formula. We thus finalize the program started by Balitsky (see Ref.~\cite{Balitsky:2004rr}) and independently by Chirilli, Kovchegov and Wertepny (see Ref.~\cite{Chirilli:2015tea}) albeit using a very different approach to carry out our calculati…
We calculate the classical single-gluon production amplitude in nucleus-nucleus collisions including the first saturation correction in one of the nuclei (the projectile) while keeping multiple-rescattering (saturation) corrections to all orders in the other nucleus (the target). In our approximation only two nucleons interact in the projectile nucleus: the single-gluon production amplitude we calculate is order-g^3 and is leading-order in the atomic number of the projectile, while resumming all order-one saturation corrections in the target nucleus. Our result is the first step towards obtain…
We calculate the inclusive gluon correlation function for arbitrary number of gluons with full rapidity and transverse momentum dependence for the initial glasma state of the p-p, p-A and A-A collisions. The formula we derive via superdiagrams generates cumulants for any number of gluons. Higher order cumulants contain information on correlations between multiple gluons, and they are necessary for calculations of higher dimensional ridges as well as flow coefficients from multi-particle correlations.
We develop a model to compute high-energy reactions of hadrons and photons. The perturbative interaction is described by gluon exchange and the non-perturbative interaction by the stochastic vacuum model which leads to quark-confinement in dipoles via a string of color fields. We study the QCD structure of the dipole-dipole scattering in momentum space focussing especially on interactions between strings. We represent the string as a collection of stringless dipoles, show confinement effects in high-energy scattering and calculate unintegrated gluon distributions of hadrons and photons. In the…
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.
